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it is well known that there is a remarkable connection between the statistical properties of certain random hermitian matrices , the zeros of the riemann zeta function , and fermionic gases @xcite . underlying each of these objects are certain one - dimensional point processes ( defined more precisely in section 2 ) whose statistical properties ( under certain limits ) are believed to be identical . the purpose of this paper is to provide generalizations of this unique point process to point processes in @xmath1-dimensional euclidean space @xmath2 for any @xmath1 , and to characterize their spatial statistics exactly . since the characterization of a point process can be viewed as the study of a system of interacting  point " particles , exact descriptions of nontrivial point processes in arbitrary space dimension , which are hard to come by @xcite , are of great value in the field of statistical mechanics . there are three prominent theories of random hermitian matrices , which model the hamiltonians of certain random dynamical systems ; see the excellent book by mehta @xcite . if the dynamical system is symmetric under time reversal , then the relevant theory for integral spin is that of the gaussian orthogonal ensemble ( goe ) or the gaussian symplectic ensemble ( gse ) for half - integer spin . on the other hand , the gaussian unitary ensemble ( gue ) models random hamiltonians without time reversal symmetry , which is relevant to certain properties of the riemann zeta function . although there are distinct one - dimensional point processes associated with each of these ensembles , our interest here will be in the gue because of its relationship to the riemann zeta function . the gue of degree @xmath14 consists of the set of all @xmath15 hermitian matrices together with a haar measure . this is the unique probability measure on the set of @xmath15 hermitian matrices that is invariant under conjugation by unitary matrices such that the individual matrix entries are independent random variables . dyson @xcite showed that the eigenvalue distrubutions of the gue are closely related to those of the  circular unitary ensemble " ( cue ) , which he exactly mapped into a problem of point particles on a unit circle interacting with a two - dimensional coulombic force law at a particular temperature . this point process on the unit circle or , equivalently , on the real line @xmath0 in the large-@xmath14 limit ( when suitably normalized ) has a pair correlation function ( defined in section 2 ) in @xmath0 at number density @xmath16 given by @xmath17 equation also applies for the gue in the limit @xmath18 such that the mean gap distance between eigenvalues at the origin is normalized ; this limit has the effect of magnifying the bulk of the eigenvalue density on @xmath0 such that is well - defined . we see that this point process is always _ negatively correlated _ , i.e. , @xmath7 never exceeds unity ( see fig . [ zeta - zero ] ) and is  repulsive " in the sense that @xmath19 as @xmath9 tends to zero . more generally , dyson @xcite proved that the @xmath3-particle correlation function ( defined in section 2 ) is given by the following determinant : @xmath20    montgomery @xcite conjectured that the pair correlation function of nontrivial zeros of the riemann zeta function ( which , according to the riemann hypothesis , lie along the critical line @xmath21 with @xmath22 ) is given exactly by the gue / cue function ( [ g2-zeros ] ) in the asymptotic limit ( high on the critical line ) when appropriately normalized . this remarkable correspondence was further established by odlyzko @xcite , who numerically verified the riemann hypothesis for the first @xmath23 nontrivial zeros of the zeta function as well as at much larger heights and confirmed that the pair correlation function agrees with ( [ g2-zeros ] ) . rudnick and sarnak @xcite proved that , under the riemann hypothesis , the nontrivial zeros have @xmath3-particle densities for any @xmath3 given by ( [ gn - zeros ] ) . the reader is referred to the excellent review article by katz and sarnak @xcite , which discusses the connection between the zeros of zeta functions and classical symmetric groups , of which the three canonical random - matrix ensembles are but special cases .    for spin - polarized free fermions in @xmath0 ( fermion gas ) at number density @xmath16 , it is known that the pair correlation function in the _ ground state _ ( i.e. , completely filling the fermi  sphere " ) is given by @xmath24 where @xmath25 is the fermi radius , which is the one - dimensional analog of the well - known three - dimensional result @xcite . therefore , we see that when @xmath26 , we obtain the gue / cue pair correlation function ( [ g2-zeros ] ) . the repulsive nature of the point process in this context arises physically from the pauli exclusion principle . very little is known about how to generalize one - dimensional point processes associated with random matrices and number - theoretic functions to higher dimensions . in this paper , we analytically obtain exact generalizations of the aforementioned one - dimensional point process in @xmath1-dimensional euclidean space @xmath2 for any @xmath1 . these processes are special cases of determinantal processes . in particular , we obtain the @xmath3-particle correlation functions for any @xmath3 , which completely specify the point processes in @xmath2 . we also show that spin - polarized fermionic systems in @xmath2 have these same @xmath3-particle correlation functions in each dimension . we show that the point processes for any @xmath1 are hyperuniform , i.e. , infinite wavelength density fluctuations vanish , and the structure factor ( or power spectrum ) @xmath4 has a nonanalytic behavior at the origin given by @xmath5 ( @xmath6 ) . the latter result implies that the pair correlation function @xmath7 tends to unity for large pair distances with a decay rate that is controlled by the power law @xmath8 . in three dimensions , such a dominant power - law decay of @xmath7 is a well - known property of bosonic systems in their ground states @xcite and , more recently , has been shown to characterize maximally random jammed sphere packings @xcite . realizations of the point processes are displayed in one and two dimensions , using a simulation technique described by us elsewhere @xcite , which vividly reveal their  repulsive " nature . in fact , we show the point processes can be characterized by an effective  hard - core " diameter that grows like the square root of @xmath1 . the nearest - neighbor distribution functions for these point processes are also studied by evaluating and rigorously bounding them . among other results , this analysis reveals that the probability of finding a large spherical cavity of radius @xmath9 in dimension @xmath1 behaves like a poisson point process but in dimension @xmath10 , i.e. , this probability is given by @xmath11 $ ] for large @xmath9 , where @xmath12 is a positive @xmath1-dependent constant . we also show that as @xmath1 increases , the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than @xmath13 . as we will see , this value of the coverage fraction has a special significance in the study of sphere packings in high dimensions .    in section 2 , we present background and definitions concerning point processes that are particularly germane to the ensuing analysis . section 3 discusses general determinantal point processes . in section 4 , we obtain the determinantal point processes in @xmath2 that are generalizations of the aforementioned one - dimensional point process associated with fermions , random matrices , and the riemann zeta function . we call the most general of these point processes  fermi - shells " point processes . the asymptotic properties of various pair statistics are investigated . we show that spin - polarized fermionic systems in @xmath2 have the same @xmath3-particle correlation functions in each dimension . section 5 analyzes various nearest - neighbor functions for the special case of the  fermi - sphere " point processes in @xmath2 . we present concluding comments in section 6 , including remarks about possible connections of our point processes to random matrix and number theory . a stochastic point process in @xmath2 is defined as a mapping from a probability space to configurations of points @xmath27 in @xmath1-dimensional euclidean space @xmath2 . more precisely , let @xmath28 denote the set of configurations such that each configuration @xmath29 is a subset of @xmath2 that satisfies two regularity conditions : ( i ) there are no multiple points ( @xmath30 if @xmath31 ) and ( ii ) each bounded subset of @xmath2 must contain only a finite number of points of @xmath32 . we denote by @xmath33 the number of points within @xmath34 , @xmath35 , where @xmath36 is the ring of bounded borel sets in @xmath2 . thus , we always have @xmath37 for @xmath35 , but the possibility @xmath38 is not excluded . we note that there exists a minimal @xmath39-algebra @xmath40 of subsets of @xmath28 that renders all of the functions @xmath33 measurable . let @xmath41 be a probability space . any measurable map @xmath42 is called a stochastic point process @xcite . henceforth , we will simply call this map a point process . note that this random setting is quite general . it incorporates cases in which the location of the points are deterministically known , such as a lattice . a point process is completely statistically characterized by specifying the countably infinite set of @xmath3-particle probability density functions @xmath43 ( @xmath44 ) @xcite . the distribution - valued function @xmath43 has a probabilistic interpretation : apart from trivial constants , it is the probability density function associated with finding @xmath3 different points at positions @xmath45 and hence has the nonnegativity property @xmath46 the point process is statistically homogeneous or translationally invariant if for every constant vector @xmath47 in @xmath2 , @xmath48 , which implies that @xmath49 where @xmath50 is the _ number density _ ( number of points per unit volume in the infinite - volume limit ) and @xmath51 is the _ @xmath3-particle correlation function _ , which depends on the relative positions @xmath52 , where @xmath53 and we have chosen the origin to be at @xmath54 . we call @xmath55 the pair correlation function .    for translationally invariant point processes without _ long - range order _ , @xmath56 when the points ( or  particles " ) are mutually far from one another , i.e. , as @xmath57 ( @xmath58 ) , @xmath59 . thus , the deviation of @xmath60 from unity provides a measure of the degree of spatial correlation between the particles with unity corresponding to no spatial correlation . note that for a translationally invariant poisson point process , @xmath60 is unity for all values of its argument . it is useful to introduce the total correlation function @xmath61 of a translationally invariant point process , which is related to the pair correlation function via @xmath62 and decays to zero for large @xmath63 in the absence of long - range order . note that @xmath64 for all @xmath65 for a translationally invariant poisson point process . an important nonnegative spectral function @xmath66 , called the structure factor ( or power spectrum ) , is defined as follows : @xmath67 where @xmath68 is the fourier transform of @xmath69 . for a translationally and rotationally invariant point process , it is useful to consider the cumulative coordination number @xmath70 , defined to be the expected number of points found in a sphere of radius @xmath9 centered at an arbitrary point of the point process , which is related to the pair correlation function as follows : @xmath71 where @xmath72 is the surface area of a @xmath1-dimensional sphere of radius @xmath9 . it is clear that since @xmath7 is a nonnegative function , @xmath70 is a monotonically increasing function of @xmath9 . the fourier transform of an @xmath73 function @xmath74 is defined by @xmath75 \rmd{{\bf r } } , \label{spec}\end{aligned}\ ] ] where @xmath73 denotes the space of absolutely integrable functions on @xmath2 . if @xmath74 is a radial function , i.e. , @xmath76 depends only on the modulus @xmath77 of the vector @xmath65 , then its fourier transform is given by @xmath78 where @xmath79 is the modulus of the wave vector @xmath80 and @xmath81 is the bessel function of order @xmath82 . the inverse transform of @xmath83 is given by @xmath84      we denote by @xmath85 the variance in the number of points @xmath86 contained within a window @xmath87 . the number variance @xmath85 for a specific choice of @xmath88 is necessarily a positive number and is generally related to the total correlation function @xmath69 for a translationally invariant point process @xcite . in the special case of a spherical window of radius @xmath89 in @xmath2 , it is explicitly given by @xmath90 ,   \label{variance}\end{aligned}\ ] ] where @xmath91 is the number variance for a spherical window of radius @xmath89 in @xmath2 and @xmath92 is the volume common to two spherical windows of radius @xmath89 whose centers are separated by a distance @xmath9 divided by @xmath93 , the volume of a spherical window of radius @xmath89 , given explicitly by @xmath94 we will call @xmath92 the _ scaled intersection volume_.    for large @xmath89 , it has been proved that @xmath91 can not grow more slowly than @xmath95 , where @xmath96 is a positive constant @xcite . we note that point processes ( translationally invariant or not ) for which @xmath91 grows more slowly than the window volume ( i.e. , as @xmath97 ) for large @xmath89 are examples of _ hyperuniform _ ( or superhomogeneous ) point patterns @xcite . this classification includes all periodic point processes @xcite , certain aperiodic point processes @xcite , one - component plasmas @xcite , point processes associated with a wide class of tilings of space @xcite , and certain disordered sphere packings @xcite . hyperuniformity implies that the structure factor @xmath66 has the following small @xmath98 behavior : @xmath99    the scaled intersection volume @xmath92 appearing in ( [ variance ] ) and its associated fourier transform will play a prominent role in this paper . the former quantity is defined by a convolution integral involving the indicator function @xmath100 for a @xmath1-dimensional spherical  window " of radius @xmath89 centered at position @xmath101 @xcite , i.e. , @xmath102 where @xmath103 is the heaviside step function @xmath104 specifically , the scaled intersection volume is given by @xmath105 the scaled intersection volume has the support @xmath106 $ ] , the range @xmath107 $ ] , and the following alternative integral representation @xcite : @xmath108 } \sin^d(\theta ) \ , \rmd\theta , \label{alpha}\end{aligned}\ ] ] where @xmath109 is the @xmath1-dimensional constant given by @xmath110}. \label{c}\end{aligned}\ ] ] torquato and stillinger @xcite found the following series representation of the scaled intersection volume @xmath92 for @xmath111 and for any @xmath1 : @xmath112 } x^{2n-1 } , \label{series}\end{aligned}\ ] ] where @xmath113 . for even dimensions , relation ( [ series ] ) is an infinite series , but for odd dimensions , the series truncates such that @xmath92 is a univariate polynomial of degree @xmath1 . in even dimensions , the scaled intersection volume involves transcendental functions ( e.g. , for @xmath114 , @xmath115 $ ] for @xmath116 ) . figure [ intersection ] provides plots of @xmath92 as a function of @xmath9 for the first five space dimensions . for any dimension , @xmath92 is a monotonically decreasing function of @xmath9 . at a fixed value of @xmath9 in the interval @xmath117 , @xmath92 is a monotonically decreasing function of the dimension @xmath1 . the fourier transform of the scaled intersection volume function ( [ alpha ] ) , which is given by @xmath118 ^ 2}{v_1(r)},\end{aligned}\ ] ] where @xmath119 is the fourier transform of the window indicator function ( [ window2 ] ) . therefore , the fourier transform of @xmath92 is the following nonnegative function of @xmath79 : @xmath120    it has been shown that finding the point process that minimizes the number variance @xmath91 is equivalent to finding the ground state of a certain repulsive pair potential with compact support @xcite . this problem is directly related to an outstanding problem in number theory involving generalized zeta functions and lattices @xcite . understanding such ground states can be facilitated by utilizing duality " relations that link ground states in real space to those in fourier space @xcite . we will be able to obtain the appropriate @xmath1-dimensional generalizations of point processes corresponding to the eigenvalues of the gue , the zeros of the riemann zeta function , or the one - dimensional fermionic gas by appealing to the notion of a determinantal point process in @xmath2 @xcite . determinantal point processes were introduced by macchi @xcite , who originally called them fermion point processes . soshnikov @xcite presented a review of this subject and discussed applications to random matrix theory , statistical mechanics , quantum mechanics , and representation theory . it is also noteworthy that examples of determinantal point processes arise in self - avoiding random walks @xcite and uniform spanning trees @xcite .    without loss of generality , the number density is set to unity ( @xmath16 ) in the ensuing discussion . let @xmath121 be a translationally invariant hermitian - symmetric kernel of an integral operator @xmath122 . a translationally invariant _ determinantal point process _ in @xmath2 exists if the the @xmath3-particle density functions for @xmath123 are given by the following determinants : @xmath124_{i , j=1,\ldots , n } , \label{rhon - det}\end{aligned}\ ] ] where @xmath125 . by virtue of the nonnegativity of the @xmath126 in the pointwise sense [ cf . ( [ positive ] ) ] and ( [ rhon - det ] ) , it follows that @xmath122 must have nonnegative minors and @xmath122 must be a nonnegative operator , which implies that @xmath127 is positive semidefinite . the latter implies that the fourier transform @xmath128 of the kernel @xmath127 is nonnegative , and this property together with the condition @xmath129 implies that @xmath130 , i.e. , @xmath131 it follows that any positive semidefinite hermitian - symmetric kernel @xmath121 whose fourier transform satisfies the inequalities in ( [ f - k ] ) describes a determinantal point process with a pair correlation function given by @xmath132 such that @xmath133 and a @xmath3-particle density given by ( [ rhon - det ] ) . we see that the total correlation function for a determinantal point process is given by @xmath134    the fact that the @xmath3-particle density functions can be written in terms of the determinant specified by ( [ rhon - det ] ) leads to bounds on @xmath126 . for example , it trivially follows that @xmath135 a less obvious but stronger upper bound is as follows : @xmath136 we remark also that : @xmath137 each of these inequalities is a consequence of the determinantal form for @xmath126 and the characteristics of @xmath138 ; in particular , follows directly from fischer s inequality @xcite and an appropriate partition of the matrix representation for the operator @xmath139 . equation is a result of the more general hadamard - fischer inequalities @xcite and the relation @xmath140 . the positive semidefinite character of @xmath139 is essential for these inequalities to hold . a unique property of determinantal point processes is that each of the @xmath3-particle correlation functions @xmath60 can be expressed completely in terms of the pair correlation function @xmath141 . namely , at unit density we may write : @xmath142 ^ 2\\ \rightarrow h(r ) = \pm\sqrt{1-g_2(r)}\label{hsqrt}.\end{aligned}\ ] ] therefore : @xmath143_{1\leq i < j\leq n}\nonumber\\   & = \det\left[\pm\sqrt{1-g_2(r_{ij})}\right]_{1\leq i < j\leq n}\label{gng2}.   \end{aligned}\ ] ] the right side of is well - defined for all @xmath144 since @xmath145 . we note that in general @xmath146 may be either positive or negative for a given value of @xmath9 as in ; therefore , the sign of the square root in must be chosen appropriately . our ability to express each @xmath60 in terms of the pair correlation function @xmath141 is a reflection of the fact that the @xmath3-particle correlation functions depend on a common kernel @xmath138 ; such a reformulation is generally not possible for an arbitrary point process . thus , one can infer the behavior of the @xmath3-particle correlation functions for @xmath147 solely from a knowledge of the behavior of @xmath141 . a trivial example of a determinantal point process is the case in which @xmath148 and @xmath149 for @xmath150 . the resulting pair correlation function is given by @xmath151 and @xmath152 , and this function belongs to the same equivalence class as the pair correlation function of the poisson point process with respect to lebesgue measure on the nonnegative reals @xmath153 ( i.e. , the functions differ only on a set of measure zero ) . note that costin and lebowitz @xcite have considered the conditions under which a pair correlation function of the form @xmath154 is a determinantal point process in @xmath2 . we note in passing that the number of points in a determinantal point process that falls in a compact set @xmath87 has the same distribution as a sum of independent bernoulli(@xmath155 ) random variables , where @xmath155 are the eigenvalues of the operator @xmath122 @xcite . moreover , hough et al . @xcite presented an algorithm to generate determinantal point process in @xmath2 , which we apply elsewhere @xcite . here we obtain the appropriate generalization of ( [ g2-zeros ] ) that corresponds to a determinantal point process in @xmath2 . first , we make the simple observation that the pair correlation function specified by ( [ g2-zeros ] ) is related to the fourier transform @xmath156 of the one - dimensional scaled intersection volume evaluated at @xmath157 and @xmath158 , namely , @xmath159 where @xmath156 is given by ( [ alpha - k ] ) for @xmath160 . a natural generalization of this pair correlation function in @xmath2 is to replace the one - dimensional fourier transform in ( [ g2-one ] ) with its @xmath1-dimensional counterpart ( [ alpha - k ] ) evaluated at @xmath157 and divided by @xmath161 , i.e. , @xmath162 however , the value of @xmath89 in each dimension must be chosen so that such a pair correlation function corresponds to a determinantal point process in that dimension . in other words , if we take the positive semidefinite function @xmath127 to be given by the following radial function : @xmath163 @xmath89 must be determined so that the conditions ( [ f - k ] ) and ( [ conditions ] ) are satisfied . noting that the expansion of @xmath164 for small @xmath9 to leading order is given by @xmath165 [ cf . ( [ conditions ] ) ] yields that @xmath166^{1/d}. \label{r}\end{aligned}\ ] ]    now we must show that a pair correlation function ( [ g2-d ] ) with @xmath167 satisfies the bounds of ( [ conditions ] ) and the bounds ( [ f - k ] ) on the spectral function @xmath168 . the square of the function @xmath146 , specified by ( [ h ] ) , is clearly positive and achieves its maximum value of unity at the origin when @xmath167 and tends to zero in the limit @xmath169 , and hence the bounds of ( [ conditions ] ) are satisfied when @xmath167 . referring to relation ( [ window - fourier ] ) for the fourier transform of the window indicator function , we see that the fourier transform of ( [ h ] ) with @xmath167 is simply the indicator function @xmath170 which automatically satisfies ( [ f - k ] ) , where @xmath100 is specified by ( [ window2 ] ) and @xmath171 is given by ( [ r ] ) .    in summary , we have demonstrated that there is a determinantal point process in @xmath2 with @xmath3-particle densities given by ( [ rhon - det ] ) with the kernel @xmath172 where @xmath171 is given by ( [ r ] ) . we will call such a determinantal point process a  fermi - sphere " point process because @xmath168 corresponds to a spherical window indicator function in fourier space and , as we will see , corresponds to the ground state of a fermionic system in which the fermi sphere is completely filled ( see section 3 ) . in particular , the pair correlation function of such a point process @xcite is given by @xmath173 moreover , the corresponding structure factor , at unit number density , takes the form @xmath174 where @xmath175 is the scaled intersection volume of two @xmath1-dimensional spheres of radius @xmath171 separated by a distance @xmath79 , i.e. , the function @xmath92 , specified by ( [ series ] ) , with the replacements @xmath176 and @xmath177 . relation ( [ s - d ] ) is easily obtained by taking the fourier transform of the total correlation function defined by ( [ h ] ) , where @xmath146 is given by ( [ h2 ] ) , and employing the definition ( [ factor ] ) for the structure factor . it follows from the properties of @xmath175 in ( [ s - d ] ) that the structure factor @xmath4 obeys the following bounds for all @xmath79 : @xmath178 and achieves its maximum value of unity for @xmath179 . the corresponding cumulative coordination number is given by @xmath180 note that the first term in ( [ z-2 ] ) is precisely the cumulative coordination number for a poisson point process in @xmath2 and that the second term is strictly negative for any @xmath181 , which is a reflection of the short - range repulsive nature of the point process . we will show that for sufficiently large @xmath9 , the dominant contribution to @xmath70 will be the poissonian term @xmath182 . we will see it is the cumulative coordination number @xmath70 , rather than the pair correlation function ( contrary to conventional wisdom ) , that enables one to identify and quantify an effective  hard - core " diameter of the determinantal point processes . it is instructive to examine the asymptotic behaviors of @xmath7 , @xmath4 , and @xmath70 for the fermi - sphere point process . by virtue of the asymptotic properties of the bessel function of arbitrary order , the small-@xmath9 and large-@xmath9 forms of the pair correlation function ( [ g2-d-2 ] ) are respectively given by @xmath183 and @xmath184 we see that @xmath7 tends to zero quadratically in @xmath9 in the limit @xmath185 , independent of the dimension . the coefficient of the quadratic term in ( [ quad ] ) for positive @xmath1 attains it maximum value of @xmath186 for @xmath160 and monotonically decreases with increasing dimension , asymptoting to @xmath187 in the limit @xmath188 . moreover , @xmath7 tends to unity for large pair distances with a decay rate that is controlled by the power law @xmath8 for any @xmath189 . the latter result implies that the structure factor @xmath4 tends to zero linearly in @xmath79 in the limit @xmath190 . using ( [ series ] ) and ( [ s - d ] ) , it is easy to verify that as @xmath79 tends to zero at @xmath16 @xmath191 where @xmath109 is a @xmath1-dependent positive constant given by ( [ c ] ) . we see that at unit density , the point process in @xmath2 for any @xmath1 is hyperuniform , and the structure factor @xmath4 has a nonanalytic behavior at the origin given by @xmath5 ( @xmath6 ) . the coefficient of the linear term in ( [ linear ] ) for positive @xmath1 attains it minimum value of @xmath192 for @xmath160 and monotonically increases with increasing dimension such that it asymptotes to @xmath193 in the limit @xmath188 . in three dimensions , this unusual linear nonanalytic behavior of the structure factor at @xmath194 is a well - known property of bosons in their grounds states @xcite and , more recently , has been shown to characterize maximally random jammed sphere packings @xcite .    in fig . [ fermions ] , we compare pair statistics in both real and fourier space for @xmath160 and @xmath195 at unit density . we see that the amplitudes of the oscillations in @xmath7 that are apparent for @xmath160 are significantly reduced in the corresponding three - dimensional pair correlation function . the smallest value of @xmath9 for which @xmath7 attains its maximum value of unity , which we denote by @xmath196 , is determined by the first positive zero of @xmath197 , which for sufficiently large @xmath1 is given by @xmath198 since @xmath171 increases with increasing @xmath1 and grows like @xmath199 for large @xmath1 , @xmath196 grows like the square root of @xmath1 for large @xmath1 . similarly , the smallest value of @xmath79 for which @xmath4 attains its maximum value of unity grows with increasing @xmath1 and for large @xmath1 grows like @xmath200 . the results immediately above enable us to obtain the following small-@xmath9 and large-@xmath9 forms of the cumulative coordination number : @xmath201\qquad ( r \rightarrow 0 ) \label{z - small}\end{aligned}\ ] ] and @xmath202 it is straightforward to prove that the asymptotic result ( [ z - large ] ) is a rigorous lower bound on @xmath70 for all @xmath9 using the identity @xmath203 , the inequality @xmath204 for all positive @xmath9 , and relation ( [ z-2 ] ) . indeed , since @xmath70 is always nonnegative , we have the lower bounds @xmath205 where the length scale @xmath206 is the zero of @xmath207 . the length scale @xmath208 can be regarded as an upper bound on the effective _ hard - core diameter _ , which clearly grows with increasing dimension as vividly illustrated in fig . [ z - plot ] , which compares the exact form of @xmath70 with the lower bound ( [ z - bounds ] ) for @xmath195 and @xmath209 . for large @xmath1 , the effective hard - core diameter is given by the asymptotic expression @xmath210 \qquad ( d \rightarrow \infty ) , \label{d - asym}\end{aligned}\ ] ] which is seen to grow like the square root of @xmath1 . this growth of the effective hard - core diameter with dimension is a conceptually important conclusion that runs counter to conventional understanding of corresponding fermionic systems , which identifies the inverse  fermi " radius , i.e. , @xmath211 $ ] ( a decreasing function of @xmath1 ) with an effective hard - core diameter . we elaborate on this point in section 3 . the hyperuniformity of the point process implies that the number variance @xmath91 for a spherical window of radius @xmath89 must grow slower than @xmath212 ( i.e. , the window volume ) for large @xmath89 . however , the fact that the large-@xmath9 behavior of the pair correlation function is controlled by the power law @xmath8 means that @xmath91 must also grow faster than the surface area of the window or @xmath213 . in particular , upon substitution of ( [ g2-d-2 ] ) into ( [ variance ] ) , an asymptotic analysis reveals that for large @xmath89 @xmath214\gamma(1+d/2)^{1/d } } \ln(r)+ c(d ) \big\}r^{d-1}\nonumber\\   + { \cal o}(r^{d-2 } ) \qquad ( r \rightarrow \infty ) , \label{fermion - var}\end{aligned}\ ] ] where @xmath215 is a @xmath1-dimensional constant of order unity . we remark that a similar asymptotic scaling is expected to hold even when the observation window is non - spherical ; a discussion of this point has been provided in @xcite . we see that the number variance scaled by the window surface area , @xmath216 , grows like @xmath217 , independent of the dimension . this unusual number variance growth law in three dimensions has also been recently seen in maximally random jammed sphere packings @xcite , which can be viewed as prototypical glasses because they are simultaneously perfectly rigid mechanically and maximally disordered . note that the coefficient multiplying @xmath217 in ( [ fermion - var ] ) decays to zero exponentially fast as @xmath218 , and , therefore , the surface - area term @xmath213 increasingly becomes the dominant one in the large-@xmath1 limit . this behavior should be contrasted with number variance for a poisson point process , which grows like the window volume , i.e. , @xmath97 . figure [ points ] graphically depicts the fermi - sphere point processes in one and two dimensions , which are generated using the algorithm of hough et . details and applications of this algorithm are reported by us elsewhere @xcite .      here we consider a generalization of the fermi - sphere point process in which @xmath219 is an indicator function for concentric rings in reciprocal space ; we denote the resulting determinantal process as the `` fermi - shells '' point process . without loss of generality , define @xmath220 radii @xmath221 , where @xmath222 and @xmath223 , such that the region of @xmath1-dimensional space within the ball @xmath224 is empty , the annulus between @xmath225 and @xmath226 is filled , and so forth . we therefore have @xmath227 filled concentric rings in reciprocal space . since the conditions for a determinantal point process are fulfilled for any indicator function in reciprocal space @xcite , the pair correlation function will still be given by . the calculation of @xmath146 proceeds as follows : @xmath228\label{apone},\end{aligned}\ ] ] which implies : @xmath229\label{aptwo},\end{aligned}\ ] ] where @xmath230 denotes the fourier transform to coordinate space . it is important to note that the values of the various @xmath221 are not independent of each other and are constrained by the density ( here set to unity ) according to : @xmath231 = ( 2\sqrt{\pi})^d \gamma(1+d/2).\end{aligned}\ ] ] the filling of fermi shells generally introduces a greater level of short - range correlations relative to the fermi - sphere case ( see fig . [ donut ] ) .      note that we do not know of any correspondence of the general fermi - shells point process in @xmath2 for any @xmath232 to random matrix theory or the zeros of any generalized zeta function in number theory . however , we can show that a spin - polarized fermionic gas in @xmath2 for any @xmath233 has the same @xmath3-particle densities . for simplicity , this comparison will be done for the fermi - sphere case , i.e. , we will demonstrate that the @xmath126 are given by ( [ rhon - det ] ) with @xmath146 given by ( [ h2 ] ) provided that the fermi radius @xmath234 . we first recall some general properties of the @xmath3-particle density functions @xmath126 for a ground - state noninteracting gas of fermions in @xmath235 . assume that we have @xmath14 spin - polarized fermions in a volume @xmath236 with number density @xmath237 . the @xmath3-particle density functions are defined for all @xmath238 with respect to the ground state @xmath239 by : @xmath240|\phi_0\rangle,\end{aligned}\ ] ] where the operators @xmath241 are the creation and annihilation field operators , respectively . using the momentum representation , the field operators are defined in terms of the particle creation and annihilation operators @xmath242 : @xmath243\\ \overline{\phi_{\mathbf{k}}(\mathbf{x } ) } = \left(\frac{1}{v}\right)^{1/2 } \exp\left[-\rmi(\mathbf{k } , \mathbf{x})\right],\end{aligned}\ ] ] where @xmath244 is the euclidean inner product of two real - valued vectors . the field operators for fermions must satisfy the following anticommutation relation : @xmath245    wick s theorem along with the anticommutation relation immediately allow us to write the expectation value in as a determinant ( equivalently , see macchi s discussion of fermion processes @xcite ) : @xmath246_{1\leq i , j \leq n}\label{eight},\end{aligned}\ ] ] where @xmath247 denotes the permutation group for @xmath3 objects . namely , wick s theorem relates the @xmath3-particle density functions to the @xmath248 ways of pairing the @xmath249 creation and annihilation field operators ; anticommutation of the operators introduces the factor of @xmath250 in .    for the ground - state system , the argument of the determinant in may be evaluated by filling the fermi sphere up to the fermi radius @xmath25 with the result : @xmath251 since @xmath252 is a translationally - invariant function , we will abuse notation slightly and write @xmath253 making the parameterization by @xmath25 explicit . in accordance with our established convention and without loss of generality , let @xmath254 . for a system of @xmath14 spin - polarized fermions , the fermi radius @xmath25 is given exactly by : @xmath255\right\}^{1/d } = k.\end{aligned}\ ] ] therefore , we may equivalently write for : @xmath256 } r^{-d/2 } j_{d/2}(rk ) = h(r).\end{aligned}\ ] ] the result in shows that our proposed fermi - sphere point process corresponds _ exactly _ to the one generated by a system of noninteracting fermions in @xmath1 dimensions . the connection for @xmath257 between this system of noninteracting fermions and the cue of random matrix theory implies that the correlations in the ground state resulting from the pauli exclusion principle are equivalent to those induced on a @xmath258 coulomb gas constrained to the unit ring and interacting via a logarithmic pair potential at finite temperature @xmath259 in units such that boltzmann s constant @xmath260 . the argument in appendix a shows that this reduction of the probability density to a classical particle system with at most two - body interactions is peculiar to the choice of the indicator function in reciprocal space ; in general , one must include at least three - body interactions to describe the system appropriately . we also mention the significance of in defining an effective hard core on the system of noninteracting fermions in each dimension . this issue is equivalent to assigning an appropriate length scale for the widely - studied ( see , e.g. , @xcite ) fermi correlation hole . an argument primarily due to slater @xcite suggests that the correlations which result from antisymmetry in the many - body wavefunction extend outward for a distance @xmath261 . the reasoning behind this choice of length scale relies on the introduction of an exchange hole into the system with approximate spherical symmetry and a radius determined by the de broglie wavelength associated with the fermi radius @xmath25 . this notion has been quantified for interacting atomic systems by considering , for example , the difference of distribution functions derived from hartree - fock and hartree wavefunctions @xcite , but the @xmath262 scaling for the noninteracting case is still reported in modern texts @xcite . however , we recall from that as @xmath1 increases , the value of the fermi radius also increases ; in fact , @xmath263 for sufficiently large @xmath1 . this behavior implies that @xmath264 as @xmath265 , meaning that the effective hard core diameter would become negligibly small for sufficiently large @xmath1 if it were to scale as @xmath262 . this conclusion contradicts the fact that @xmath19 pointwise as @xmath265 for all @xmath266 . in other words , the range in @xmath9 over which one finds a small probability of finding two particles in close proximity increases with dimension , requiring a different means by which to quantify the extent of the effective hard core . the definition of @xmath70 in suggests instead that we take the value of @xmath208 in as a measure of the effective hard core diameter . we note that since @xmath70 is a nonnegative monotonically increasing function of @xmath9 and , therefore , nonzero for some range in @xmath9 over which its lower bound is zero , @xmath208 must represent an upper bound to the effective hard core radius . this representation of the effective hard - core diameter is quantitatively well - defined for any dimension due to the inclusion of the @xmath267 factor under the integral in from the surface area of the @xmath1-dimensional ball ; it is this factor which appropriately rescales @xmath141 such that represents a true lower bound on @xmath70 in any dimension by the argument above . it is also for this reason that neither @xmath7 nor @xmath4 alone are sufficient to define the effective hard - core diameter in a quantitative manner . in contrast to slater s scaling of the effective hard core diameter , @xmath268 in any dimension by the definition of the fermi radius , which is in accordance with the high - dimensional behavior of both @xmath141 and @xmath25 . it is useful to characterize point processes by examining other quantities besides the @xmath3-particle correlation functions . one popular descriptor used in one dimension is the so - called  gap " distribution function @xmath269 @xcite , which characterizes the spacing between the points . in the random matrix theory literature , this quantity often has erroneously and misleadingly been called the  nearest - neighbor - spacing " distribution because gaps to the right of some reference point are considered . however , @xmath269 makes no distinction between gaps to the left or right of some reference point . the quantity @xmath270 gives the probability of finding a gap ( a line interval empty of points ) of length between @xmath271 and @xmath272 . the function @xmath269 is called the _ chord length probability density _ in the theory of random media @xcite .    in the case of random matrix theory , there exist exact representations of @xmath269 for the spacings of the eigenvalues in the goe , gue , and gse , but they can only be determined numerically for general situations . a remarkably accurate approximation for the goe in the infinitely - large matrix limit is the so - called wigner surmise . the wigner surmise has been generalized to any of the aforementioned ensembles as follows : @xmath273 where the parameters @xmath274 and @xmath275 , which depend on reciprocal temperature @xmath276 , are obtained from the normalization of both @xmath269 and its first moment , or the average gap size @xmath277 . for the goe , gue , and gse , @xmath278 , and 4 , respectively , and @xmath279 , @xmath280 , @xmath281 , @xmath282 , and @xmath283 , @xmath284 .    for @xmath232 , the gap distribution function is strictly not a meaningful descriptor of point processes . the natural generalizations of @xmath269 in higher dimensions are the nearest - neighbor functions @xcite . nearest - neighbor functions describe the probability of finding the _ nearest point _ of a point process in @xmath2 at some given distance from a _ reference point _ in the space . such statistical quantities are called  void " or  particle " nearest - neighbor functions if the the reference point is an arbitrary point of the space or an actual point of the point process , respectively .      first , we recall the definitions of the void and particle nearest - neighbor probability density functions @xmath285 and @xmath286 , respectively : @xmath287 @xmath288 it is useful to introduce the associated dimensionless  exclusion " probabilities @xmath289 and @xmath290 defined as follows : @xmath291 @xmath292    it follows that the exclusion probabilities are _ complementary cumulative distribution functions _ associated with the density functions and thus are related to the latter via @xmath293 and @xmath294 differentiating the exclusion - probability relations with respect to @xmath9 gives @xmath295 and @xmath296 the @xmath3th moment of @xmath297 is defined as @xmath298 of particular interest to us is the _ mean nearest - neighbor distance _ it is useful to express the density functions @xmath300 and @xmath297 as a product of two functions as follows : @xmath301 and @xmath302 where @xmath303 is the surface area of a @xmath1-dimensional sphere of radius @xmath9 given by ( [ area - sph ] ) . the quantities @xmath304 and @xmath305 are called the  conditional " nearest - neighbor functions and have the following interpretation : @xmath306 @xmath307 thus , it follows that the exclusion probabilities are also given by @xmath308 \label{ev - gv}\end{aligned}\ ] ] and @xmath309 . \label{ep - gp}\end{aligned}\ ] ] we remark that knowledge of any one function @xmath138 , @xmath310 , or @xmath252 ( either void or particle ) is sufficient to determine the other two functions via the relations mentioned above .      the nearest - neighbor functions can be expressed as infinite series whose terms are integrals over the @xmath3-particle density functions @xcite . for example , the void and particle exclusion probability functions for a translationally invariant point process are respectively given by @xmath311 and @xmath312 the corresponding series for @xmath300 and @xmath297 are obtained from the series above using ( [ derv - ev ] ) and ( [ derv - ep ] ) . in general , an exact evaluation of the aforementioned infinite series are not possible , except for simple processes such as the poisson point process . in the latter instance , because @xmath313 , both series ( [ ev - series ] ) and ( [ ep - series ] ) can be summed exactly to give @xmath314 , \label{e - poisson}\end{aligned}\ ] ] where @xmath315 is the volume of a @xmath1-dimensional sphere of radius @xmath9 given by ( [ v1 ] ) . therefore , for a poisson point process , we have from ( [ derv - ev ] ) , ( [ derv - ep ] ) , ( [ def - hv2 ] ) , and ( [ def - hp2 ] ) that @xmath316 , \qquad g_v(r)=g_p(r)=1.\end{aligned}\ ] ] we see that there is no distinction between the void and particle quantities for the poisson distribution , which is generally not the case for correlated point processes .      torquato has given rigorous upper and lower bounds on the so - called _ canonical @xmath3-point correlation function _ @xmath317 ( with @xmath318 and @xmath319 ) for point processes in @xmath2 . since the void and particle exclusion probabilities and nearest - neighbor probability density functions are just special cases of @xmath320 , then we also have strict bounds on them for such models . let @xmath28 represent either @xmath321 , or @xmath322 and @xmath323 represent the @xmath79th term of the series for these functions . furthermore , let @xmath324 be the partial sum . then it follows that for any of the exclusion probabilities or nearest - neighbor probability density functions , we have the bounds @xmath325    application of the aforementioned inequalities yield the first three successive bounds on the nonnegative void exclusion probability : @xmath326 where @xmath327 is the intersection volume of two @xmath1-dimensional spheres ( cf . ) . the corresponding first two nontrivial bounds on the nonnegative nearest - neighbor probability density function @xmath300 are as follows : @xmath328 where @xmath329 is the surface area of the intersection volume @xmath330 . bounds on the conditional function @xmath304 follow by combining the bounds above on @xmath331 and @xmath300 and definition ( [ def - hv2 ] ) . for example , we obtain the following bounds @xmath332 and @xmath333 which should only be applied for @xmath9 such that @xmath304 remains positive . the bounds above lead to the following properties of the nearest - neighbor functions at the origin : @xmath334    similarly , the first three successive bounds on the particle exclusion probability are given by @xmath335 where @xmath70 is the cumulative coordination number defined by ( [ z ] ) . the corresponding first two nontrivial bounds on the nonnegative nearest - neighbor probability density function @xmath297 are as follows : @xmath336 where @xmath337 is the radial dirac delta function . bounds on the conditional function @xmath305 follow by combining the bounds on @xmath338 and @xmath297 and definition ( [ def - hp2 ] ) . for example , we obtain the following bounds : @xmath339 and @xmath340 which should only be applied for @xmath9 such that @xmath305 remains positive . the bounds above lead to the following properties of the nearest - neighbor functions at the origin : @xmath341    we now obtain bounds on the mean nearest - neighbor distance @xmath342 at unit density using the aforementioned upper and lower bounds on @xmath338 . let us define the following distances : @xmath343 \rmd r \label{l1}\end{aligned}\ ] ] and @xmath344 \rmd r , \label{l2}\end{aligned}\ ] ] where @xmath196 is the location of the zero of @xmath345 . in light of the bounds ( [ ep - bound1 ] ) and ( [ ep - gp - bound ] ) , it is clear that @xmath346 and @xmath347 bound @xmath342 from below and above , respectively , i.e. , @xmath348        the fermi - sphere point process is unique in that both the particle and void exclusion probabilities may be expressed as determinants over @xmath15 matrices , the elements of which are related to overlap integrals of the basis functions @xmath349 $ ] on @xmath350 , a @xmath1-dimensional ball of radius @xmath9 centered at the origin ( the exact location of the ball s center is irrelevant since the point process is translationally invariant ) . the thermodynamic limit can then be taken appropriately . we provide the details of this analysis elsewhere @xcite and immediately state the results : @xmath351\label{det1}\\ e_p(r ) = e_v(r)\tr\{a[\mathbb{i}-m(r)]^{-1}\}\label{det2},\end{aligned}\ ] ] where @xmath352 is the @xmath15 identity matrix , and the matrices @xmath353 and @xmath88 are defined by : @xmath354 where @xmath50 is the number density . we recall that knowledge of @xmath331 and @xmath338 is sufficient to determine all of the remaining nearest - neighbor functions @xmath355 and @xmath356 . note that as @xmath357 , @xmath358 for all @xmath359 and @xmath360 , which provides the necessary results @xmath361 and @xmath362 we mention that these determinants must be evaluated numerically but essentially exactly for finite @xmath15 matrices , where @xmath14 is chosen to be sufficiently large to capture accurately the behavior of the system in the thermodynamic limit . evidence for this convergence is provided in another paper @xcite . we now obtain bounds for nearest - neighbor functions for the fermi - sphere point process and compare them to the corresponding aforementioned exact results . using the identity @xmath363 ^ 2,\end{aligned}\ ] ] the leading order term of the small-@xmath9 expansion ( [ quad ] ) , and the upper bound ( [ ev - bound2 ] ) , we obtain the weaker upper bound @xmath364 ^ 2,\end{aligned}\ ] ] which is exact through terms of order @xmath365 . therefore , we also have @xmath366 ^ 2   + { \cal o}(r^{2d+3 } ) \end{aligned}\ ] ] and @xmath367 we see that through order @xmath368 , @xmath369 . this bound will be shown to apply for any @xmath9 .    employing the inequality @xmath370 [ cf . ( [ rhon - ineq ] ) ] in the upper bound ( [ ep - bound2 ] ) on @xmath331 and lower bound ( [ hp - bound2 ] ) on @xmath297 , we find the following corresponding weaker bounds : @xmath371 and @xmath372 these bounds in conjunction with the analogous evaluations of the bounds ( [ gp - bound1 ] ) and ( [ gp - bound2 ] ) on the conditional pair function @xmath305 yields its exact small-@xmath9 behavior up through terms of order @xmath373 : @xmath374 comparing this expansion to ( [ quad ] ) reveals that @xmath375 up through terms of order @xmath373 . we can also show that @xmath376 and @xmath377 these results are obtained from definitions ( [ ev - gv ] ) and ( [ ep - gp ] ) and the following upper bounds on the exclusion probabilities : @xmath378 \qquad \mbox{for all } \quad r \label{ev - gv - bound}\end{aligned}\ ] ] and @xmath379 \qquad \mbox{for all } \quad r. \label{ep - gp - bound}\end{aligned}\ ] ] to prove and , we recall the series representations of @xmath380 and @xmath381 in and , respectively , which we may rewrite in the following more compact form : @xmath382 it is important to note that the series in converge absolutely for all @xmath144 , which is easily seen from the inequalities @xmath383^k}{k!}\label{evkineq}\\ e_p^{(k ) } \leq \frac{\rho^k}{k ! } \int_{\mathbb{r}^d } \prod_{j=2}^{k+1 } g_2(r_{1j})\theta(r - r_{1j } ) \rmd\mathbf{r}_j = \frac{[z(r)]^k}{k ! } \label{epkineq}.\end{aligned}\ ] ] equations and follow directly from the inequalities in and , respectively . it is therefore true that @xmath384 as @xmath385 . we now wish to compare the series in with the following series representations of the proposed upper limits in and : @xmath386^k}{k!}\label{evulseries}\\ e_p^{(ul)}(r ) = 1-z(r)+\sum_{k=2}^{+\infty } \frac{[-z(r)]^k}{k!}\label{epulseries}.\end{aligned}\ ] ] note that the series in , , and agree up to their second terms ; the alternating series test then implies that the contributions from the remaining terms is no greater than the magnitude of the third terms in the series . equations and therefore show that the series in and in actuality do bound the series in from above , thereby proving the claims . the lower bounds in and immediately follow from monotonicity and positivity of the exponential function along with and . the left panel of fig . [ ev - ep-4d ] compares the upper bound ( [ ev - gv - bound ] ) and lower bound ( [ ev - bound1 ] ) of @xmath380 for a fermi - sphere point process for @xmath387 to the corresponding exact evaluation . the right panel of the same figure compares the upper and lower bounds on @xmath338 for @xmath387 as obtained from ( [ ep - bound1 ] ) and ( [ ep - gp - bound ] ) , respectively , to the corresponding exact evaluation of it . the upper bounds on both exclusion probabilities provide reasonable estimates of the exact results as compared to the corresponding lower bounds . figure [ gp-4d ] compares upper and lower bounds on @xmath305 for @xmath387 as obtained from ( [ ep - bound1 ] ) and ( [ ep - gp - bound ] ) , respectively , to the corresponding exact evaluation of it . not surprisingly , the bounds become better estimators as the dimension increases and therefore can be profitably used in high dimensions , where exact evaluations are difficult to obtain . the left panel of fig . [ ev - ep-17d ] shows the upper bound ( [ ev - gv - bound ] ) and lower bound ( [ ev - bound1 ] ) for a fermi - sphere point process for @xmath209 . it is seen that the bounds essentially converge to unity for the range @xmath388 and are relatively close to one another for @xmath389 . the right panel of fig . [ ev - ep-17d ] depicts the analogous bounds on @xmath338 for @xmath209 . note that the bounds on @xmath331 behave almost exactly like the bounds on @xmath338 at this value of @xmath1 . this graphically suggests that as @xmath1 becomes large , the exact expressions for @xmath331 and @xmath338 approach the same step function , which we demonstrate below . we now show that both @xmath381 and @xmath380 tend to the same step function @xmath390 in the large-@xmath1 limit , where @xmath208 is the estimate of the effective hard - core diameter defined by ( [ fermihole ] ) . we begin by utilizing the following upper bound on @xmath338 : @xmath391 , \quad r\ge d. \end{array}\right . \label{ep - bound3}\end{aligned}\ ] ] this upper bound is obtained by combining lower bound ( [ z - bounds ] ) on the cumulative coordination number @xmath70 and upper bound ( [ ep - gp - bound ] ) , and hence is a weaker upper bound on @xmath338 than ( [ ep - gp - bound ] ) . as @xmath1 tends to infinity , we see that the upper bound ( [ ep - bound3 ] ) tends to the step function @xmath390 , i.e. , @xmath392 now we show that @xmath381 is bounded from below by the same step function in this limit , i.e. , @xmath393 to prove this , we note that the the lower bound ( [ ep - bound1 ] ) on @xmath338 tends to a unit step function as @xmath1 becomes large [ cf . [ ev - ep-17d ] ] whose discontinuity location can not exceed the zero of @xmath345 , @xmath394 , which can be estimated to be given by @xmath395^{1/(d+2)}.\end{aligned}\ ] ] this estimate , which bounds the zero from below and becomes increasingly accurate as @xmath1 tends to infinity , is obtained by substituting the leading term of the asymptotic expansion ( [ z - small ] ) into @xmath396 and solving for @xmath394 . for large @xmath1 , @xmath394 has the asymptotic expansion @xmath397 \qquad ( d \rightarrow \infty ) , \label{d0-asym}\end{aligned}\ ] ] comparison of this expansion to that of the corresponding one for @xmath208 [ cf . ( [ d - asym ] ) ] reveals that @xmath208 bounds @xmath394 from above for sufficiently large @xmath1 and @xmath398 in the limit @xmath218 . thus , the lower bound on @xmath338 in this limit is given by ( [ ep - lower ] ) . combination of this lower bound with upper bound ( [ ep - upper ] ) leads to the following high - dimensional behavior : @xmath399 where we recall that @xmath208 grows like @xmath200 for large @xmath1 [ cf . ( [ d - asym ] ) ] . following the analogous analysis using the lower bound ( [ ev - bound1 ] ) and upper bound ( [ ev - gv - bound ] ) on @xmath331 , we can show @xmath400    it is noteworthy that this analysis means that the lower bound ( [ z - bounds ] ) on the cumulative coordination number @xmath70 becomes exact in the limit @xmath218 . this in turn implies an  effective " pair correlation function @xmath401 that tends to the following step function as @xmath1 tends to infinity : @xmath402 this effective pair correlation function @xmath403 is to be distinguished from the true pair correlation function ( [ g2-d ] ) , which tends to unity for distances beyond the length scale @xmath208 , but remains a quadratic function of @xmath9 for small @xmath9 . because the effective pair correlation function is based on the behavior of @xmath70 , which weights the true @xmath7 by @xmath404 ( due to the appearance of @xmath303 , the surface area of a sphere of radius @xmath9 ) , @xmath403 tends to a step function in the high - dimensional limit . the fact that the oscillations of @xmath7 seen in low dimensions ( cf . [ fermions ] ) effectively vanish in the large-@xmath1 limit is consistent with the decorrelation principle @xcite , which , roughly speaking , states that unconstrained correlations that exist in low dimensions vanish as @xmath1 tends to infinity , and all higher - order correlation functions @xmath60 for @xmath147 may be expressed in terms of @xmath141 within some small error . we have already shown in that the latter claim is true for _ any _ determinantal point process . the former claim is seen from the form of @xmath405 in , which immediately suggests that asymptotic unconstrained correlations in the fermi sphere point process diminish with respect to increasing dimension @xmath1 . in other words , @xmath141 flattens at unity for sufficiently large @xmath9 as @xmath1 becomes large , which implies that long - range correlations between any two particles in the system diminish with respect to increasing dimension , leaving only the small-@xmath9 correlations , which extend outward for a greater range in @xmath9 as @xmath1 increases . this conclusion in conjunction with implies that for sufficiently large @xmath1 and for large particle separations , @xmath406 where @xmath352 is the @xmath407 identity matrix . therefore , all @xmath3-particle correlations also diminish for large particle separations and large @xmath1 in accordance with a decorrelation of the system . we remark that the fact that @xmath19 as @xmath357 does not affect the statement of the decorrelation principle for the fermi sphere point process ; borrowing the language of quantum mechanics , these correlations arise from the constraint of antisymmetry in the many - particle wavefunction and therefore must be enforced in any dimension .    for sufficiently large @xmath1 this analysis implies that the system reduces to a sphere packing with an effective hard - core diameter equal to @xmath208 . the connection to sphere packings implies that the fraction of space @xmath408 covered by the spheres at unit number density is bounded from above by the following inequality @xmath409 interestingly , minkowski proved a lower bound on the coverage fraction of the densest lattice sphere packings that asymptotically is controlled by @xmath13 @xcite . we remark on the significance of this result in section 6 . we now obtain analytical estimates of the mean nearest - neighbor distance @xmath342 at unit density using the general upper and lower bounds on @xmath342 [ cf . ( [ l1 ] ) and ( [ l2 ] ) ] . consider the arithmetic average of ( [ l1 ] ) and ( [ l2 ] ) : @xmath410 for low dimensions , the arithmetic average of the upper and lower bounds for the fermi - sphere point process provide reasonable estimates of @xmath342 , as seen in table [ mean - table ] for the first four space dimensions , which also includes the corresponding exact results . we see that the estimate @xmath411 captures the nonmonotone dependence of the mean nearest - neighbor distance with dimension . moreover , the table shows that the upper bound becomes the dominant contribution to @xmath412 as @xmath1 increases and @xmath411 becomes increasingly accurate as the space dimensions increases . c c c c c c @xmath1 & upper bound&lower bound & average of bounds & exact & @xmath208 + 1 & 0.917808 & 0.658199 & 0.788003 & 0.725728 & 0.5 + 2 & 0.688071 & 0.581193 & 0.634632 & 0.649823 & 0.564190 + 3 & 0.670304 & 0.593975 & 0.632139 & 0.654511 & 0.620350 + 4 & 0.687631 & 0.625009 & 0.656320 & 0.679561 & 0.670938 +    [ mean - table ]    note use of the upper bound ( [ ep - bound3 ] ) enables us to obtain the following weaker but analytically solvable upper bound on @xmath342 : @xmath413 \rmd r \nonumber\\ & = d\left [ 1+\frac{\gamma(1/d,1 ) e}{d}\right ] , \label{lambda - bound}\end{aligned}\ ] ] where @xmath414 is the incomplete gamma function . for large @xmath1 , we have the asymptotic expression @xmath415 \nonumber \\ & = d\left[1 + \frac{0.5963473622\ldots}{d } + { \cal o}\left(\frac{1}{d^2}\right)\right ] \qquad ( d \rightarrow \infty).\end{aligned}\ ] ] moreover , using the lower bound @xmath416 \rmd r   \label{lambda - bound2}\end{aligned}\ ] ] where @xmath394 is the zero of @xmath396 , and the aforementioned asymptotic analysis of the lower bound on @xmath338 , yields @xmath417    in summary , combination of the bounds ( [ lambda - bound ] ) and ( [ lambda - bound2 ] ) and the asymptotic expression ( [ d0-asym ] ) , enables us to conclude that the mean nearest - neighbor distance approaches the length scale @xmath208 as @xmath1 becomes large , i.e. , @xmath418 which asymptotically grows like the square root of @xmath1 according to ( [ d - asym ] ) and , as we concluded above , specifies the location of the step - function discontinuity of @xmath338 , @xmath331 and @xmath419 in the large-@xmath1 limit . table [ mean - table ] shows that the length scale @xmath208 is already an accurate estimate of the mean nearest - neighbor distance for @xmath387 . it is noteworthy that the asymptotic mean nearest - neighbor - distance formula ( [ lambda - asym ] ) is precisely the same as the asymptotic form of the mean nearest - neighbor distance of a poisson point process . the latter for any dimension at unit density is given by @xmath420 @xcite , which in the high - dimensional limit is exactly equal to @xmath208 . the fact that the mean nearest - neighbor distance for the fermi - sphere point process behaves like that of a poisson point process in the high - dimensional limit is not surprising in light of the decorrelation principle @xcite . note that the expression for the mean nearest - neighbor distance @xmath421 for any density @xmath50 can be related to the corresponding quantity @xmath422 at unit density by the simple scaling relation @xmath423 figure [ mean ] shows the mean nearest - neighbor distance as a function of density for various dimensions ; the cases @xmath160 and @xmath387 are exact evaluations and the instance @xmath209 is obtained from the upper bound prediction ( [ lambda - bound ] ) and the scaling relation ( [ scaling ] ) , which is expected to be a highly accurate estimate . we conclude this section by making some remarks about the conditional nearest - neighbor functions @xmath304 and @xmath305 . the fact that the exclusion probabilities @xmath331 and @xmath338 tend to the same step function in the high - dimensional limit implies that @xmath304 and @xmath305 have the same large-@xmath9 behavior as @xmath1 tends to infinity . in fact , our exact evaluations of @xmath304 and @xmath305 for a finite range of @xmath9 in low dimensions indicate that each function becomes linear in @xmath9 for large @xmath9 and the ratio @xmath424 tends to unity . figure [ gv - gp-1d-4d ] shows our evaluations of both @xmath304 and @xmath305 for the first four space dimensions for the range @xmath425 ( see our companion paper @xcite for further numerical details ) .    in one dimension , one can show that the nearest - neighbor void functions are directly related to a radial function that is a solution to a second - order nonlinear differential equation @xcite . this differential equation can be evaluated exactly for small and large @xmath9 . in particular , using this asymptotic analysis leads to the following large-@xmath9 behavior for @xmath304 for the one - dimensional fermi - sphere point process : @xmath426 the coefficient @xmath427 can be compared to the corresponding wigner surmise prediction of @xmath428 , which is obtained via the one - dimensional relation that exactly links @xmath331 to the gap distribution function @xmath269 @xcite , @xmath429 and eq . ( [ wigner ] ) with @xmath430 . presumably , @xmath305 has the same asymptotic behavior for @xmath160 , a conclusion supported by our numerical results @xcite . interestingly , for both conditional probability functions to have the same linear behavior for large @xmath9 implies that the corresponding exclusion probability functions have the following large-@xmath9 behavior for finite @xmath1 : @xmath431 \qquad ( r \rightarrow \infty ) , \label{ev - poisson}\end{aligned}\ ] ] where @xmath12 is a positive @xmath1-dependent constant . thus , this analysis reveals that the probability of finding a large spherical cavity of radius @xmath9 for a fermi - sphere point process in dimension @xmath1 behaves similar to that of a poisson point process but in dimension @xmath10 . for a poisson point process , the constant @xmath432 $ ] , [ cf . ( [ e - poisson ] ) and therefore if @xmath331 for a fermi - sphere point process in @xmath2 for large @xmath9 behaved exactly like that for a poisson point process in @xmath433 , then @xmath304 would be given by @xmath434 } r + { \cal o}(1)\qquad ( r \rightarrow \infty ) . \label{gv - poisson}\end{aligned}\ ] ] for the first four space dimensions , the coefficients multiplying @xmath9 in ( [ gv - poisson ] ) are given by @xmath435 , 2 , @xmath436 , and @xmath437 , respectively . this can be compared to the exact coefficients , which for @xmath438 and 4 are given by @xmath439 , @xmath440 , and @xmath441 , respectively . the last three values are estimates that we have obtained based upon extrapolations from the evaluations of @xmath304 ( see ref . @xcite for further numerical details ) . we see that the probability of finding a large spherical cavity of radius @xmath9 in @xmath2 is approximately the same as that for a poisson point process in @xmath433 , implying linear growth of @xmath304 or @xmath305 for large @xmath9 . it is not unreasoanble to conclude that this approximation becomes increasingly accurate as @xmath1 increases . further justification for this remarkable behavior is given in section 6 . we have obtained and characterized a new class of determinantal point processes in @xmath2 , the most general of which we call fermi - shells point processes . the @xmath3-particle correlation functions for any @xmath3 and @xmath1 , which completely characterize the point process , are determined analytically . we focused primarily on a special case , the fermi - sphere point process , which in one dimension is identical to the point process that characterizes the spacings of the eigenvalues of the gue ( as well as the cue ) , the conjectured spacings of the nontrivial zeros of the riemann zeta function , and the positions of spin - polarized fermions in their ground state ( i.e. , completely filling the fermi  sphere " ) . we are not aware of any correspondence of the general fermi - shells point process in @xmath2 for any @xmath232 to random matrix theory or the zeros of any generalized zeta function in number theory . if our determinantal point processes have connections to random matrices in arbitrary space dimension , the latter must be non - hermitian . for example , ginibre @xcite showed that there are two - dimensional determinantal point processes that correspond to complex eigenvalues of random non - hermitian matrices . thus , it is an open question whether the fermi - shells point process has any correspondence to random matrices . we analyzed in great detail properties of pair statistics , including the pair correlation function , the structure factor , and cumulative coordination number , as a function of spatial dimension @xmath1 . the point processes for any @xmath1 are shown to be hyperuniform such that the structure factor ( or power spectrum ) @xmath4 has a nonanalytic behavior at the origin given by @xmath5 ( @xmath6 ) . the latter result implies that the pair correlation @xmath7 tends to unity for large pair distances with a decay rate that is controlled by the power law @xmath8 . in three dimensions , such a dominant power - law decay of @xmath7 is a well - known property of bosonic systems in their ground states @xcite and , more recently , has been shown to characterize maximally random jammed sphere packings @xcite . we also graphically displayed one- and two - dimensional realizations of the point processes in order to vividly reveal their  repulsive " nature and demonstrated that they can be characterized by an effective  hard - core " diameter that grows like the square root of @xmath1 . our study of the nearest - neighbor functions of the fermi - sphere point process resulted in some noteworthy conclusions . for example , we have seen that the probability of finding a large spherical cavity of radius @xmath9 in @xmath2 is approximately the same as that for a poisson point process in @xmath433 , implying linear growth of @xmath304 or @xmath305 for large @xmath9 . this is a remarkable result because it represents the first class of nontrivial point processes that we are aware of whose conditional nearest - neighbor functions do not asymptote to a constant ; see refs . @xcite and @xcite for examples of correlated equilibrium and nonequilibrium point processes that possess @xmath304 or @xmath305 with constant asymptotes , respectively . it is clear that the unusual property of linear growth of @xmath442 or @xmath443 for large @xmath9 is due to the long - range nature of the repulsive interactions . since a fermi - sphere point process in @xmath2 is always hyperuniform , i.e. , large wavelength density fluctuations vanish , the probability of finding a large spherical cavity must be smaller than the corresponding quantity for any point process that is not hyperuniform , such as a poisson point process in @xmath2 . the probability of finding a large spherical cavity is of course smaller in a poisson point process in dimension @xmath10 compared to that in dimension @xmath1 . moreover , it is easy to show that the asymptotic form ( [ ev - poisson ] ) for @xmath331 or @xmath338 for large @xmath9 is always between the aforementioned corresponding rigorous upper and lower bounds on the exclusion probabilities . we also found that the fermi - sphere point process becomes a sphere packing in the high - dimensional limit with an effective hard - core diameter equal to the length scale @xmath208 [ cf . ( [ fermihole ] ) ] . thus , the fraction of space @xmath408 covered by the spheres at unit number density is bounded from above by @xmath13 . this coverage fraction has a special significance in the study of sphere packings ; it arises not only in minkowski s famous century - old lower bound on the density of the densest lattice sphere packings @xcite but in lower bounds for saturated and disordered packings @xcite as well as the highest achievable density in the  ghost " random sequential addition packing @xcite . it should be noted , however , that there is strong evidence that there exist disordered sphere packings with @xmath408 not only greater than @xmath13 @xcite but with densities that exponentially improve on minkowski s lower bound @xcite .    elsewhere @xcite we report results on the extremes of the nearest - neighbor statistics as well as voronoi statistics of the fermi - sphere point processes in the first four space dimensions . in other work , we will quantify clustering and percolation properties of fermi - sphere point processes . the authors are grateful to peter sarnak and juan maldacena for useful discussions . this work was supported by the national science foundation under grant no . our purpose here is to show that the @xmath3-particle probability density function for an arbitrary determinantal point process , even in one dimension , can not be written as a boltzmann factor of @xmath14 classical particles interacting through one- and two - body potentials at a finite temperature . although this claim is true for each of the canonical ensembles of random - matrix theory , we show via a counterexample in one dimension that intrinsic @xmath3-particle interactions with @xmath444 are generally necessary to describe the system . it is relatively straightforward to express the @xmath3-particle probability density function for the cue as a boltzmann factor of a classical system of pairwise - interacting particles . in analogy with the formalism of the one - dimensional fermi - sphere point process introduced in section 4.1 , we fill the fermi line at a constant density . thus , may write @xcite : @xmath445_{n , m}=\prod_{n < m}[\exp(\rmi x_n)-\exp(\rmi x_m)];\end{aligned}\ ] ] therefore , @xmath446_{n , m}\right|^2&=\exp\left[-2\sum_{n < m}\ln|\exp(\rmi x_n)-   \exp(\rmi x_m)|\right]\\ & = \exp\left[-2\sum_{n < m}\ln|\sin(x_n - x_m)/2|\right]\label{a3}.\end{aligned}\ ] ] for small eigenvalue separations @xmath447 , the result in reduces to the form : @xmath448 ; i.e. , the state @xmath449 is skipped . we may then express the three - body form of the probability density as : @xmath450_{n , m}\right|^2=\frac{64}{(2\pi)^3 } [ 3 + 2    \cos(x - y)+2 \cos(x - z)+2 \cos(y - z)]\nonumber\\   \times   \sin\left(\frac{x - y}{2}\right)^2 \sin\left(\frac{x - z}{2}\right)^2    \sin\left(\frac{y - z}{2}\right)^2,\end{aligned}\ ] ] where we have used @xmath451 and @xmath271 to represent @xmath452 and @xmath453 ; the subscripts denote the state @xmath3 of each particle . the last three factors have the form of the pair interaction in . to check whether the pre - factor containing cosines can be written in the same form , we write : @xmath454\label{threepot}.\end{aligned}\ ] ] assume that @xmath455 . it must then be true that : @xmath456\\ & = \frac{\ln(3)}{3 } + \frac{1}{2 } \ln[5 + 4 \cos(x - y)],\end{aligned}\ ] ] but by substituting this expression into , we see that we do not recover the original functional form . therefore , we have shown by contradiction that for general determinantal point processes , the interaction potential must contain at least intrinsic three - body terms . 20            odlyzko a m , _ on the distribution of spacings between zeros of the zeta functions _ , 1987 _ math . comput . _ * 48 * 273 ; odlyzko a m , _ the @xmath457-nd zero of the riemann zeta function _ , in : van frankenhuysen m and lapidus m l , 2001 _ dynamical , spectral , and arithmetic zeta functions _ ( providence : ams ) pp . 139 - 144                                        a sphere packing is a point process in which there is a minimal positive distance between any pair of points . according to ref . @xcite , a _ disordered _ sphere packing in @xmath2 is one in which the pair correlation function @xmath459 decays to its long - range value of unity faster than @xmath460 for some @xmath461 .
it is well known that one can map certain properties of random matrices , fermionic gases , and zeros of the riemann zeta function to a unique point process on the real line @xmath0 . here 
 we analytically provide exact generalizations of such a point process in @xmath1-dimensional euclidean space @xmath2 for any @xmath1 , which are special cases of determinantal processes . 
 in particular , we obtain the @xmath3-particle correlation functions for any @xmath3 , which completely specify the point processes in @xmath2 . 
 we also demonstrate that spin - polarized fermionic systems in @xmath2 have these same @xmath3-particle correlation functions in each dimension . 
 the point processes for any @xmath1 are shown to be hyperuniform , i.e. , infinite wavelength density fluctuations vanish , and the structure factor ( or power spectrum ) @xmath4 has a nonanalytic behavior at the origin given by @xmath5 ( @xmath6 ) . 
 the latter result implies that the pair correlation function @xmath7 tends to unity for large pair distances with a decay rate that is controlled by the power law @xmath8 , which is a well - known property of bosonic ground states and more recently has been shown to characterize maximally random jammed sphere packings . 
 we graphically display one- and two - dimensional realizations of the point processes in order to vividly reveal their  repulsive " nature . 
 indeed , we show that the point processes can be characterized by an effective  hard - core " diameter that grows like the square root of @xmath1 . 
 the nearest - neighbor distribution functions for these point processes are also evaluated and rigorously bounded . among other results 
 , this analysis reveals that the probability of finding a large spherical cavity of radius @xmath9 in dimension @xmath1 behaves like a poisson point process but in dimension @xmath10 , i.e. , this probability is given by @xmath11 $ ] for large @xmath9 and finite @xmath1 , where @xmath12 is a positive @xmath1-dependent constant . 
 we also show that as @xmath1 increases , the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than @xmath13 . 
 this coverage fraction has a special significance in the study of sphere packings in high - dimensional euclidean spaces . 
 _ keywords _ : correlation functions , point processes , fermions , random matrix theory , number theory