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this article deals with the relationship between zeros and critical points of random polynomials in one complex variable . recall that a _ critical point _ of a polynomial @xmath10 is a root of its derivative @xmath11 there are many results concerning the location of critical points of polynomials whose roots are known . one of the most famous examples is the gauss lucas theorem , which offers a geometric connection between the roots of a polynomial and the roots of its derivative . [ thm : gauss ] if @xmath10 is a non - constant polynomial with complex coefficients , then all zeros of @xmath11 belong to the convex hull of the set of zeros of @xmath10 . there are many refinements of theorem [ thm : gauss ] ; we refer the reader to @xcite and references therein .    a probabilistic version of the problem was first studied by pemantle and rivin @xcite . specifically , pemantle and rivin raised the following question . for a random polynomial @xmath10 , when are the zeros of @xmath11 stochastically similar to the zeros of @xmath10 . before introducing their results , we fix the following notation . for a polynomial @xmath10 of degree @xmath1 , we define the empirical measure constructed from the roots of @xmath10 as @xmath12 where each root in the sum is counted with multiplicity and @xmath13 is the unit point mass at @xmath14 . in particular , when @xmath10 is a random polynomial , @xmath15 becomes a random probability measure . for the critical points of @xmath10 , we introduce the notation @xmath16 in other words , @xmath17 is the empirical measure constructed from the critical points of @xmath10 . let @xmath18 be independent and identically distributed ( iid ) random variables taking values in @xmath19 , and let @xmath2 be their common probability distribution . for each @xmath20 , consider the polynomial @xmath21 under the assumption that @xmath2 has finite one - dimensional energy , pemantle and rivin @xcite show that @xmath22 converges weakly to @xmath2 as @xmath1 tends to infinity . let us recall what it means for a sequence of random probability measures to converge weakly . let @xmath23 be a topological space ( such as @xmath24 or @xmath19 ) , and let @xmath25 be its borel @xmath26-field . let @xmath27 be a sequence of random probability measures on @xmath28 , and let @xmath2 be a probability measure on @xmath28 . @xmath29 converges weakly to @xmath2 in probability _ as @xmath30 ( and write @xmath31 in probability ) if for all bounded continuous @xmath32 and any @xmath33 , @xmath34 in other words , @xmath31 in probability as @xmath30 if and only if @xmath35 in probability for all bounded continuous @xmath36 . similarly , we say _ @xmath29 converges weakly to @xmath2 almost surely _ as @xmath30 ( and write @xmath31 almost surely ) if for all bounded continuous @xmath32 , @xmath37 almost surely . kabluchko @xcite generalized the results of pemantle and rivin to the following . [ thm : pmk ] let @xmath2 be an arbitrary probability measure on @xmath19 , and let @xmath38 be a sequence of iid random variables with distribution @xmath2 . for each @xmath20 , let @xmath0 be the degree @xmath1 polynomial given in . then @xmath22 converges weakly to @xmath2 in probability as @xmath30 . subramanian , in @xcite , verified a special case of theorem [ thm : pmk ] when @xmath2 is supported on the unit circle in the complex plane . naturally , one may ask whether the assumptions in theorem [ thm : pmk ] ( such as the roots @xmath38 being independent ) can be relaxed . in @xcite , the first author managed to prove a version of theorem [ thm : pmk ] for random polynomials with dependent roots provided the roots lie on the unit circle and satisfy a number of technical conditions . in particular , the results in @xcite apply to characteristic polynomials of random unitary matrices and other matrices from the classical compact groups ( the eigenvalues of such matrices are known to not be independent ) . similar results for characteristic polynomials of nearly hermitian matrices were studied in ( * ? ? ? * section 2.5 ) . in @xcite , reddy considers polynomials whose zeros are chosen randomly from two deterministic sequences of complex numbers in which the empirical measures for both sequences converge to the same limit . it is shown that the limiting empirical measure of the zeros and critical points agree for these polynomials , yielding a version of theorem [ thm : pmk ] where the randomness can be reduced and independence still remains . however , as the following example shows , the randomness in theorem [ thm : pmk ] can not be completely eliminated ( i.e. , the theorem does not always hold for sequences of deterministic polynomials ) . let @xmath39 . then the roots of @xmath0 are the @xmath1-th roots of unity , and so @xmath40 converges weakly to the uniform measure on the unit circle as @xmath1 tends to infinity . however , all @xmath41 critical points of @xmath0 are located at the origin . hence , @xmath42 for all @xmath1 . we use asymptotic notation ( such as @xmath43 ) under the assumption that @xmath30 . in particular , @xmath44 , @xmath45 , @xmath46 , and @xmath47 denote the estimate @xmath48 , for some constant @xmath49 independent of @xmath1 and for all @xmath50 . if we need the constant @xmath51 to depend on another constant , e.g. @xmath52 , we indicate this with subscripts , e.g. @xmath53 , @xmath54 , @xmath55 , and @xmath56 . we write @xmath57 if @xmath58 for some @xmath59 that goes to zero as @xmath30 . specifically , @xmath60 denotes a term which tends to zero as @xmath30 . to introduce our results , we first consider a special case of the polynomial @xmath0 , defined in , when @xmath2 is the uniform probability measure on the unit circle centered at the origin . in this case , theorem [ thm : pmk ] implies that @xmath61 converges weakly in probability to @xmath2 as @xmath30 . a numerical simulation of this result is shown in figure [ fig : iidcirc ] ; as can be seen , all critical points of @xmath62 lie very close to the unit circle . on the other hand , if we consider the polynomial @xmath63 for some deterministic point @xmath3 outside the unit circle , we see in figure [ fig : iidcircdetout ] that one of the critical points leaves the unit disk and lies very close to @xmath3 . however , the remaining critical points still lie close to the unit circle . the goal of this note is to describe the pairing between the root @xmath3 and the nearby critical point . more generally , we consider the case when several deterministic zeros are appended to the random polynomial @xmath0 and when @xmath2 is an arbitrary measure in the complex plane with compact support ( not just the uniform distribution on the unit circle ) . see , for example , figures [ fig : iidcirc3detout ] and [ fig : distinguished : iidblob ] . let us mention that this pairing phenomenon between roots and critical points has been observed previously for random polynomials . hanin @xcite proves a similar pairing result when a number of deterministic roots are appended to a random polynomial whose roots are chosen independently from a probability measure @xmath2 on the riemann sphere . hanin s proof is guided by an intuitive electrostatic interpretation of the zeros and critical points . unlike the results in @xcite however , our results do not require the measure @xmath2 to have bounded density or require the deterministic roots to satisfy a separation condition . in addition , our methods are significantly different than those used in @xcite and allow us to describe the exact number of critical points lying in a region outside the support of @xmath2 . in a separate paper @xcite , hanin considers the joint distribution of roots and critical points for a class of gaussian random polynomials . however , the polynomials considered in @xcite are quite different than the model considered in this paper . finally , let us mention the work of dennis and hannay @xcite from the physics literature , which gives an electrostatic explanation for the pairing of critical points and zeros of random polynomials and characteristic polynomials of random matrices .      to begin , we first consider the analogue of theorem [ thm : pmk ] when @xmath64 deterministic zeros are appended to the random polynomial @xmath0 in . [ thm : genkabluchko ] let @xmath2 be an arbitrary probability measure on @xmath19 , and suppose @xmath38 are iid random variables with distribution @xmath2 . for each @xmath20 , let @xmath65 be a deterministic non - negative integer no larger than @xmath1 such that @xmath66 . in addition , let @xmath67 be a deterministic triangular array of complex values , and let @xmath68 then @xmath61 converges weakly to @xmath2 in probability as @xmath30 . theorem [ thm : genkabluchko ] is a generalization of theorem [ thm : pmk ] . indeed , theorem [ thm : pmk ] can be recovered from theorem [ thm : genkabluchko ] by taking @xmath69 . unsurprisingly , we prove theorem [ thm : genkabluchko ] in appendix [ sec : genkabluchko ] by slightly generalizing the methods developed by kabluchko in @xcite . let us discuss the intuition behind theorem [ thm : genkabluchko ] . to do so , we must begin with theorem [ thm : pmk ] . roughly speaking , theorem [ thm : pmk ] describes the phenomenon that if @xmath0 is a degree @xmath1 random polynomial , then @xmath70 in probability as @xmath30 . in other words , the limiting behavior of the critical points is the same as the limiting behavior of the roots . while theorem [ thm : pmk ] only applies to random polynomials with iid roots , the same phenomenon has been observed for other ensembles of random polynomials @xcite , and numerical simulations show that it should be true for many other models . stated another way , the behavior in appears to be universal among random polynomials . let us now consider the polynomial @xmath0 from theorem [ thm : genkabluchko ] . it follows from the law of large numbers that @xmath71 weakly almost surely as @xmath30 since @xmath72 . therefore , if the convergence in applies to the polynomial @xmath0 , the triangle inequality would immediately imply that @xmath61 also converges weakly to @xmath2 in probability . this heuristic is the basis for our proof of theorem [ thm : genkabluchko ] . the above heuristic also hints that the condition @xmath66 in theorem [ thm : genkabluchko ] is sharp . indeed , if @xmath73 deterministic roots were to be appended , the limiting distribution is , in general , not @xmath2 as shown by the following example . let @xmath74 and @xmath75 . define @xmath76 where @xmath38 are iid random variables uniformly distributed on the unit circle centered at the origin in the complex plane . then , by theorem [ thm : pmk ] , @xmath22 converges weakly to the uniform measure on the unit circle in probability as @xmath30 . however , the polynomial @xmath77 has at least @xmath78 critical points at the origin . in particular , @xmath79 for @xmath1 sufficiently large . among other things , this implies that @xmath80 does not converge weakly to the uniform probability measure on the unit circle as @xmath30 . while theorem [ thm : genkabluchko ] shows that the global behavior of the critical points is unchanged by the addition of @xmath64 deterministic roots , the addition of one or more deterministic roots can create a number of outlying critical points as illustrated in figures [ fig : iidcircdetout ] and [ fig : iidcirc3detout ] . one way of viewing this phenomenon is to view the deterministic roots as a small perturbation of the original polynomial . this small perturbation is not enough to change the global distribution of the critical points ; it may , however , as observed in the figures above , create a small number of outlying critical points . our main results below describe these outliers .      before we consider the perturbed model , we first consider the case when there are no deterministic roots . in this initial case , we want to determine exactly where the critical points of the random polynomial @xmath0 , defined in , are located . this way , when we do append the small perturbation of deterministic roots , we will be able to tell exactly what effect the perturbation has had . let @xmath2 be a probability measure on @xmath19 , and suppose @xmath81 are iid random variables with distribution @xmath2 . in view of the gauss lucas theorem ( theorem [ thm : gauss ] ) , the roots of @xmath82 , must lie in @xmath83 , the convex hull of the support of @xmath2 . however , as we discussed above in the case when @xmath2 is supported on the unit circle ( shown in figure [ fig : iidcirc ] ) , nearly all of the critical points appear near the support of @xmath2 , which is only a small subset of the convex hull . thus , our goal is to determine the exact subset of @xmath83 where the critical points will lie , with high probability . we do so in the theorem below . to define this set where the critical points are located , we will first need to introduce the cauchy stieltjes transform . let @xmath2 be a probability measure on @xmath19 , and let @xmath84 be the cauchy stieltjes transform of @xmath2 defined by @xmath85 also , define @xmath86 to be the set of zeros of @xmath84 . if @xmath2 has compact support , it turns out that @xmath87 ; see proposition [ prop : mmu ] for details . for @xmath33 , we also define the set @xmath88 to be the @xmath89-neighborhood of @xmath90 . here , @xmath91 is the distance from @xmath92 to a set @xmath93 . the following theorem shows that all critical points of @xmath0 must lie inside @xmath94 with high probability . [ thm : nooutliers ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath81 are iid random variables with distribution @xmath2 . then , for every @xmath33 , there exists @xmath95 ( depending only on @xmath2 and @xmath89 ) such that , with probability at least @xmath96 , the polynomial @xmath97 has no critical points outside @xmath98 .    by the gauss lucas theorem ( theorem [ thm : gauss ] ) , the critical points of @xmath0 must lie inside @xmath83 . thus , theorem [ thm : nooutliers ] actually reveals that , with high probability , @xmath0 has no critical points outside @xmath99 . we now justify our choice of the set @xmath94 as the correct location of the critical points . first , in the case that @xmath2 is degenerate , @xmath100 for some @xmath101 , which has critical point @xmath102 with multiplicity @xmath41 . this example shows that clearly the critical points of @xmath0 may lie in @xmath103 . the next example shows that the critical points can also be in a neighborhood of the zero set @xmath104 . let @xmath105 for some @xmath106 with @xmath107 and @xmath108 , and assume @xmath38 are iid random variables with distribution @xmath2 . then @xmath109 for some non - negative integers @xmath110 with @xmath111 . almost surely , for @xmath1 sufficiently large , @xmath112 , and , in this case , @xmath113 thus , by the law of large numbers , @xmath0 has a critical point at @xmath114 almost surely . on the other hand , @xmath115 has exactly one zero located at @xmath116 .    by the borel cantelli lemma , theorem [ thm : nooutliers ] immediately implies the following corollary . let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath38 are iid random variables with distribution @xmath2 . fix @xmath33 . then , almost surely , for @xmath1 sufficiently large , the polynomial @xmath97 has no critical points outside @xmath98 . we now consider the outlying critical points depicted in the figures above . to do so , we will need the following notation . for a polynomial @xmath117 of degree @xmath1 , we let @xmath118 be the critical points of @xmath117 counted with multiplicity .    [ thm : simple ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath38 are iid random variables with distribution @xmath2 . let @xmath119 , and assume @xmath7 are deterministic complex numbers ( which do not depend on @xmath1 ) ; in addition , suppose there are @xmath120 values @xmath121 not in @xmath90 . then , there exists @xmath122 such that the following holds for any fixed @xmath123 . almost surely , for @xmath1 sufficiently large , there are exactly @xmath120 critical points ( counted with multiplicity ) of the polynomial @xmath124 outside @xmath98 , and after labeling these critical points correctly , @xmath125 for each @xmath126 . theorem [ thm : simple ] describes exactly the phenomenon we observe in the figures above . in particular , this theorem shows that each deterministic root outside @xmath90 creates one outlying critical point , which is asymptotically close to the deterministic root .    in figure [ fig : iidcircdetin ] , we give the results of a numerical simulation when @xmath2 is the uniform probability measure on the unit circle centered at the origin , @xmath127 , and @xmath128 . as predicted by theorem [ thm : simple ] , the figure shows that there is exactly one critical point near the outlier @xmath129 . for comparison , we provide the following example which shows that theorem [ thm : simple ] fails for deterministic polynomials . let @xmath130 and @xmath131 . then the roots of @xmath9 are @xmath132-th roots of unity with an outlier at @xmath133 . however , we will show that @xmath9 has no critical points near @xmath133 . indeed , @xmath134 and so the critical points are the solutions of @xmath135 for @xmath136 , we have @xmath137 for @xmath1 sufficiently large . this implies that @xmath138 for every @xmath92 with @xmath136 . hence , for @xmath1 sufficiently large , there are no critical points of @xmath9 in the disk @xmath139 . more generally , this argument shows that for a fixed @xmath140 , there are no critical points of @xmath9 in the disk @xmath141 for sufficiently large @xmath1 . we next state two generalizations of theorem [ thm : simple ] . both results deal with the case when the deterministic points @xmath7 ( as well as the integer @xmath142 ) are allowed to depend on @xmath1 . because the points can now depend on @xmath1 , some additional technical assumptions are required . these technical assumptions are trivially satisfied when @xmath7 do not depend on @xmath1 . as such , theorem [ thm : simple ] is actually a corollary of the following more general result . [ thm : main ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath38 are iid random variables with distribution @xmath2 . for each @xmath20 , let @xmath67 be a triangular array of deterministic complex numbers with @xmath143 , and assume @xmath144 fix @xmath33 , and suppose that for all sufficiently large @xmath1 , there are no values of @xmath67 in @xmath145 and there are @xmath120 values @xmath146 outside @xmath147 . then , almost surely , for @xmath1 sufficiently large , there are exactly @xmath120 critical points ( counted with multiplicity ) of the polynomial @xmath148 outside @xmath149 , and after labeling these critical points correctly , @xmath150 for each @xmath126 . the @xmath151-magnitude assumption in is required for our proof . however , we conjecture that this condition is not needed . in fact , in the case when @xmath152 , we can remove this assumption , and we obtain the following stronger result . [ thm : distinguished ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath38 are iid random variables with distribution @xmath2 . for each @xmath20 , let @xmath67 be a triangular array of deterministic complex numbers with @xmath143 . fix @xmath33 , and suppose that for all sufficiently large @xmath1 , there are no values of @xmath67 in @xmath145 and there is one value @xmath153 outside @xmath147 . then , almost surely , for @xmath1 sufficiently large , there is exactly one critical point of the polynomial @xmath148 outside @xmath149 , and after labeling the critical points correctly , @xmath154    if @xmath155 , then implies that , almost surely , @xmath156 more generally , if @xmath157 , theorem [ thm : distinguished ] yields that , almost surely , @xmath158 in other words , the location of the outlying critical point @xmath159 is asymptotically close to the outlying root @xmath153 . we present a numerical simulation of theorem [ thm : distinguished ] in figure [ fig : distinguished : iidblob ] . the rest of the paper is devoted to the proof of our main results . in section [ sec : tools ] , we develop several tools we will need for the proofs . the proof of theorem [ thm : nooutliers ] is presented in section [ sec : proof : nooutliers ] . we prove theorems [ thm : simple ] , [ thm : main ] , and [ thm : distinguished ] in section [ sec : proofs ] . finally , the proof of theorem [ thm : genkabluchko ] is given in appendix [ sec : genkabluchko ] . we present here some tools we will need to prove our main results .      we will need the following complex - valued version of hoeffding s inequality . [ lemma : hoeffding ] let @xmath160 be iid complex - valued random variables which satisfy @xmath161 almost surely for some @xmath162 . then there exist absolute constants @xmath163 such that @xmath164 \right| \geq t \right ) \leq c \exp \left ( - c n t^2 / k^2 \right)\ ] ] for every @xmath165 . let @xmath166.\ ] ] if @xmath167 , then @xmath168 or @xmath169 . so , we have @xmath170 the claim now follows from the classic ( real - valued ) version of hoeffding s inequality ( see @xcite ) since @xmath171 and @xmath172 . we introduce @xmath173-nets as a convenient way to discretize a compact set . let @xmath174 be a subset of @xmath19 , and @xmath33 . a subset @xmath175 of @xmath174 is called an @xmath89-net of @xmath174 if every point @xmath176 can be approximated within @xmath89 by some point @xmath177 , i.e. so that @xmath178 .    for a finite set @xmath175 , we let @xmath179 denote the cardinality of @xmath175 . we will need the following estimate for the size of an @xmath89-net . [ lemma : net ] let @xmath180 be a compact subset of @xmath181 for some @xmath182 . then , for every @xmath33 , there is an @xmath89-net @xmath175 of @xmath180 such that @xmath183    let @xmath184 be a maximal @xmath185-separated subset of @xmath186 . in other words , @xmath184 is such that @xmath187 for all @xmath188 with @xmath189 , and no subset of @xmath190 containing @xmath184 has this property . such a set can always be constructed by starting with an arbitrary point in @xmath190 and at each step selecting a point that is at least @xmath185 distance away from those already selected . since @xmath190 is compact , this procedure will terminate after a finite number of steps . the maximality property implies that @xmath184 is an @xmath185-net of @xmath190 . indeed , otherwise there would exist @xmath191 that is at least @xmath185-far from all points in @xmath184 . so @xmath192 would still be an @xmath185-separated set , contradicting the maximality property above . moreover , the separation property implies that the balls of radii @xmath193 centered at the points in @xmath184 are disjoint . in addition , all such balls lie in the ball of radius @xmath194 centered at the origin . comparing areas gives @xmath195 and hence @xmath196 we now use @xmath184 to construct an @xmath89-net of @xmath180 . indeed , we construct @xmath175 iteratively using the following procedure . let @xmath197 be an enumeration of the points in @xmath184 , and set @xmath198 . given @xmath199 for @xmath200 , we construct @xmath201 as follows :    1 . if the ball of radius @xmath185 centered at @xmath202 does not intersect @xmath180 , then let @xmath203 . 2 .   if the ball of radius @xmath185 centered at @xmath202 does intersect @xmath180 , let @xmath204 be an element of the intersection and set @xmath205 .    now take @xmath206 . by the procedure above , it follows that @xmath207 . it remains to show that @xmath175 is an @xmath89-net of @xmath180 . let @xmath208 . since @xmath209 , there exists @xmath210 such that @xmath211 . this means that the ball of radius @xmath185 centered at @xmath212 intersects @xmath180 . thus , from the procedure above , there exists @xmath177 such that @xmath213 . therefore , by the triangle inequality , @xmath214 . we will need the following companion matrix result , which describes a matrix whose eigenvalues are the critical points of a given polynomial . this result appears to have originally been developed in @xcite ( see ( * ? ? ? * lemma 5.7 ) ) . however , the same result was later rediscovered and significantly generalized by cheung and ng @xcite . [ thm : companion ] let @xmath215 for some complex numbers @xmath216 , and let @xmath180 be the diagonal matrix @xmath217 . then @xmath218 where @xmath219 is the @xmath220 identity matrix and @xmath221 is the @xmath220 all - one matrix . theorem [ thm : companion ] allows us to translate the problem of studying critical points to a problem involving the eigenvalues of certain matrices . for studying the eigenvalues of such matrices , we will need the following lemmata . [ lemma : block ] suppose @xmath222 , and @xmath180 are matrices of dimension @xmath220 , @xmath223 , @xmath224 and @xmath225 , respectively . if @xmath226 is invertible , then @xmath227    the conclusion follows immediately from the decomposition @xmath228 where @xmath229 and @xmath230 are the identity matrices of dimension @xmath220 and @xmath225 , respectively . a similar proof is given in ( * ? ? * section 0.8.5 ) . [ lemma : sm ] suppose @xmath226 is an invertible matrix and @xmath231 are column vectors . if @xmath232 , then @xmath233    lemma [ lemma : sm ] can be found in @xcite ; see also ( * ? ? ? * section 0.7.4 ) for a more general version of this identity known as the sherman  morrison woodbury formula . we will also require the following bound involving the difference of two determinants . for a matrix @xmath226 , we let @xmath234 denote the spectral norm of @xmath226 , i.e. ,  @xmath234 is the largest singular value of @xmath226 . [ lemma : detdiff ] let @xmath226 and @xmath235 be @xmath236 matrices . if @xmath237 , then @xmath238    by the leibniz formula for the determinant , it follows that @xmath239 where the sums range over all permutations @xmath26 of @xmath240 and @xmath241 is the sign of the permutation @xmath26 . we now take advantage of the fact that the spectral norm of a matrix bounds the magnitude of each entry . in particular , @xmath242 and @xmath243 thus , by multiple applications of the triangle inequality , we obtain @xmath244 uniformly in @xmath26 . combining this bound with completes the proof . we collect here some additional tools and facts we will need . first , we note that if @xmath2 has compact support , then the convex hull of the support of @xmath2 is also a compact set ; see ( * ? ? ? * corollary 5.33 ) for details . the following proposition shows that the zero set of the cauchy  stieltjes transform of @xmath2 must lie inside the convex hull of the support of @xmath2 . [ prop : mmu ] let @xmath2 be a probability measure on @xmath19 with compact support . if @xmath245 for some @xmath246 , then @xmath247 . let @xmath248 , and define @xmath249 suppose @xmath250 . then @xmath251 for any @xmath252 . since @xmath103 is compact , it follows from ( * ? ? ? * corollary 5.33 ) that @xmath253 is also compact . thus , by the hyperplane separation theorem , there exists a pair of parallel lines , separated by a gap @xmath33 , separating @xmath253 and @xmath254 . let @xmath255 be the angle these lines make with the real axis ( if they do not meet the real axis take @xmath256 ) . then @xmath257 is of the same sign for all @xmath258 and @xmath259 for all @xmath258 . thus , we obtain @xmath260 as @xmath103 is compact , there exists @xmath261 such that @xmath262 for all @xmath258 . hence , we conclude that @xmath263 and the proof is complete . we will also need the following observation concerning the translation of roots and critical points . [ prop : translate ] let @xmath117 be a monic polynomial of degree @xmath1 , and suppose @xmath264 are the critical points of @xmath117 counted with multiplicity . then , for any @xmath101 , the critical points of @xmath265 are @xmath266 .    since @xmath117 is a monic polynomial of degree @xmath1 , @xmath267 thus , @xmath268 and the claim follows . this section is devoted to the proof of theorem [ thm : nooutliers ] . for @xmath33 , define @xmath269 to be the @xmath89-neighborhood of the support of @xmath2 . we begin with the following concentration inequality . [ lemma : lln ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath81 are iid random variables with distribution @xmath2 . then , for every @xmath270 , @xmath271 for some absolute constants @xmath272 . let @xmath270 , and define @xmath273 we assume @xmath180 is nonempty as the conclusion is trivial otherwise . let @xmath175 be an @xmath274-net of @xmath180 . by lemma [ lemma : net ] , @xmath175 can be chosen so that @xmath275    we observe that @xmath276 almost surely for every @xmath277 . thus , almost surely , for @xmath208 , @xmath278 hence , for @xmath279 , @xmath280 in other words , the function @xmath281 is almost surely lipschitz continuous on @xmath180 with lipschitz constant @xmath282 . similarly , for @xmath283 , @xmath284    suppose @xmath285 . as @xmath286 and @xmath84 are both continuous on the compact set @xmath180 , there exists @xmath208 such that @xmath287 . since @xmath175 is an @xmath274-net of @xmath180 , there exists @xmath288 such that @xmath289 . so , by the reverse triangle inequality and the fact that the @xmath286 and @xmath84 are lipschitz continuous , we have @xmath290 therefore , by the union bound , we conclude that @xmath291 as @xmath292 for @xmath208 , hoeffding s inequality ( lemma [ lemma : hoeffding ] ) and the bound in imply that @xmath293 for some absolute constants @xmath95 . thus , combining , , and yields @xmath294 as desired . let @xmath33 . with probability one , @xmath276 for each @xmath277 . thus , the zeros of @xmath295 outside of @xmath94 are exactly the critical points of @xmath0 outside of @xmath94 . we will show that @xmath296 has no zeros in @xmath297 . the claim then follows immediately since , by the gauss lucas theorem ( theorem [ thm : gauss ] ) , all the critical points of @xmath0 lie in @xmath83 . since @xmath2 has compact support , @xmath83 is also a compact set ( see ( * ? ? ? * corollary 5.33 ) ) , and hence @xmath180 is compact . as @xmath84 is a continuous function on @xmath180 , @xmath298 achieves its minimum on @xmath180 , which , by definition of @xmath94 can not be zero ( since @xmath94 contains the zero set @xmath299 ) . thus , there exists @xmath300 such that @xmath301    since @xmath180 is compact , there exists @xmath261 ( depending only on @xmath103 ) such that @xmath302 for all @xmath208 . thus , by lemma [ lemma : lln ] ( taking @xmath303 ) , we obtain @xmath304 for some absolute constants @xmath305 . hence , on the complementary event , we have @xmath306 for all @xmath208 . since the constants @xmath307 and @xmath308 only depend on @xmath2 and @xmath89 , the proof is complete . the proof of theorem [ thm : genkabluchko ] presented here is modeled after kabluchko s proof of ( * ? ? ? * theorem 1.1 ) . we note that theorem [ thm : genkabluchko ] does not follow from the results in @xcite , and the notable difference between our proof and the one given in @xcite is that we must control the additional contribution coming from the deterministic triangular array . for convenience , we use @xmath29 and @xmath419 to mean @xmath40 and @xmath420 , respectively and define @xmath421 to be the collection of values present in the deterministic triangular array . we let @xmath422 represent lebesgue measure on @xmath423 , and we denote the positive and negative parts of the real logarithm by @xmath424 for @xmath425 . we use the convention that @xmath426 so that @xmath427 is a function taking values in the extended real line .      under the same hypotheses as in theorem [ thm : genkabluchko ] and with the additional assumption that there is a set @xmath428 of lebesgue measure zero for which @xmath429 implies @xmath430 it follows that @xmath431 converges weakly to @xmath2 in probability as @xmath30 . [ thm : kabl : detcond ]    unfortunately , we can not always guarantee that the deterministic array satisfies condition . to get around this issue , we will work on subsequences where the condition does hold ; specifically , the proof of theorem [ thm : genkabluchko ] will require the following corollary of theorem [ thm : kabl : detcond ] . assume the same hypotheses as in theorem [ thm : genkabluchko ] and , in addition , suppose @xmath432 is a subsequence of @xmath29 for which there is a set @xmath433 of zero lebesgue measure such that @xmath429 implies @xmath434 then @xmath435 converges weakly to @xmath2 in probability as @xmath30 . [ cor : kabl : subseq ]    we show that @xmath432 is a subsequence of a new sequence of random measures ( modified from @xmath29 ) for which condition does hold . to this end , define the sequence @xmath436 by @xmath437 and the random polynomial @xmath438 also let @xmath439 and @xmath440 denote @xmath441 and @xmath442 , respectively . by construction , @xmath432 and @xmath435 are subsequences of @xmath439 and @xmath440 , respectively . now , @xmath443 , and for @xmath429 , @xmath444 thus , theorem [ thm : kabl : detcond ] implies that @xmath440 converges weakly to @xmath2 in probability as @xmath30 . it follows that the subsequence @xmath435 also converges to @xmath2 weakly in probability as @xmath445 . let @xmath29 be a sequence of random probability measures on @xmath423 , and suppose @xmath2 is a deterministic probability measure on @xmath423 . then , @xmath29 converges weakly to @xmath2 in probability if and only if each subsequence of @xmath29 contains a further subsequence that converges weakly to @xmath2 in probability . [ lma : kabl : subcriterion ]    observe that , for each bounded and continuous function @xmath446 , the sequence @xmath447 is a sequence of complex - valued random variables whose subsequences are of the form @xmath448 , where @xmath432 is a subsequence of @xmath29 . in addition , @xmath449 is a constant . thus , the claim follows by applying theorem 2.6 on page 20 of @xcite to the random variables @xmath447 . we now prove theorem [ thm : genkabluchko ] by way of corollary [ cor : kabl : subseq ] and lemma [ lma : kabl : subcriterion ] . the proof of theorem [ thm : kabl : detcond ] is delayed until section [ sec : kabl : proofdetcond ] . fix a subsequence @xmath435 of @xmath419 . we will show that there exists a further subsequence that converges weakly to @xmath2 in probability , which , by lemma [ lma : kabl : subcriterion ] would complete the proof of theorem [ thm : genkabluchko ] . clearly , @xmath432 is a subsequence of @xmath29 . if @xmath422 denotes lebesgue measure on @xmath423 , then markov s inequality implies that , for any @xmath33 , @xmath450 the last expression tends to zero as @xmath445 by the local integrability of the logarithm and the fact that @xmath66 . thus , the sequence of functions @xmath451converges to zero in measure as @xmath445 . among other things , this implies that there exists a subsequence of this sequence that converges to zero for almost every @xmath452 ( see , for instance , theorem 2.30 on page 61 of @xcite for details ) . let @xmath453 denote the corresponding subsequence of random measures . by corollary [ cor : kabl : subseq ] , we have that @xmath454 converges weakly to @xmath2 in probability as @xmath455 , completing the proof . it remains to prove theorem [ thm : kabl : detcond ] . the proof presented here is modeled after the arguments given in @xcite . the case where @xmath2 is degenerate is straightforward to establish by computing @xmath431 explicitly and directly verifying that @xmath456 almost surely as @xmath30 for any bounded and continuous function @xmath446 . we now consider the case that @xmath2 is non - degenerate . the proof of theorem [ thm : kabl : detcond ] will reduce to studying the logarithmic derivative @xmath457 of @xmath0 defined by the formula @xmath458 specifically , theorem [ thm : kabl : detcond ] will follow from lemma [ lma : kabl : intsmall ] below . we also now state a related lemma ( lemma [ lma : kabl : lnsmall ] ) , which we will need later . note that these two lemmas are very similar to ( * ? ? ? * lemmas 2.1 and 2.2 ) ; however , neither lemma follows directly from the results in @xcite because of the deterministic contribution to @xmath457 .      under the assumptions of theorem [ thm : kabl : detcond ] , for any continuous , compactly supported function @xmath462 , we have @xmath463 in probability as @xmath30 . ( recall that @xmath422 denotes lebesgue measure on @xmath19 . ) [ lma : kabl : intsmall ]    we now prove theorem [ thm : kabl : detcond ] assuming lemma [ lma : kabl : intsmall ] . the key idea is the following formula ( see , for instance , ( * ? ? ? * section 2.4.1 ) ) , which relates the integral in to the measures @xmath29 and @xmath419 . for any polynomial @xmath10 that is not identically zero , @xmath464 in the distributional sense , where each root in the sum is counted with multiplicity . in other words , for any compactly supported , smooth function @xmath462 , we have @xmath465 from this relationship we obtain that , for any smooth , compactly supported function @xmath462 , @xmath466 in view of lemma [ lma : kabl : intsmall ] , the integral on the right tends to zero in probability as @xmath467 . in addition , by the law of large numbers and the fact that @xmath66 , @xmath468 almost surely as @xmath30 . hence , for any smooth , compactly supported function @xmath462 @xmath469 in probability as @xmath30 . since @xmath2 is a probability measure , we conclude from a simple approximation argument that @xmath419 converges weakly to @xmath2 in probability . this completes the proof of theorem [ thm : kabl : detcond ] . we now turn our attention to proving lemmas [ lma : kabl : lnsmall ] and [ lma : kabl : intsmall ] . we begin with lemma [ lma : kabl : lnsmall ] , which we will need to prove lemma [ lma : kabl : intsmall ] . first , we construct the exceptional set @xmath470 described in lemma [ lma : kabl : lnsmall ] from several smaller subsets . the first of these , @xmath471 , contains points where @xmath2 misbehaves , while another , @xmath472 , includes values too close to the deterministic array . define the set @xmath471 by @xmath473 @xmath471 has lebesgue measure zero since @xmath474 by the fubini tonelli theorem . we now construct the subset @xmath472 by applying the borel  cantelli lemma . recall that the set @xmath475 , defined in , is at most countable , and hence @xmath476 . thus , for a fixed @xmath477 and @xmath478 , @xmath479 by markov s inequality , where @xmath49 is an absolute constant equal to the integral of @xmath480 over @xmath423 . thus , we obtain @xmath481 since @xmath482 . it follows by the borel cantelli lemma and the fact that @xmath475 is countable that there exists a set @xmath483 of lebesgue measure zero such that , for every @xmath484 , @xmath485 for all but finitely many pairs @xmath486 . we conclude that , for @xmath487 , @xmath488 where the asymptotic notation @xmath489 means the implicit constant is allowed to depend on @xmath14 .    if we define @xmath470 to be @xmath490 then @xmath470 has lebesgue measure zero and , as we shall see , satisfies the requirements of lemma [ lma : kabl : lnsmall ] . ( recall the definition of @xmath428 from the statement of theorem [ thm : kabl : detcond ] above . ) notice that @xmath470 contains the atoms of @xmath2 and the values in the deterministic triangular array . fix @xmath460 , and let @xmath33 be given . by markov s inequality , for any @xmath477 , we have @xmath492}{{\varepsilon}^2 n^2}\\ & = \frac{1}{{\varepsilon}^2 n^2 } \int_{\mathbb{c}}\log^2_{-}{\left\vertz - y\right\vert}\,d\mu(y)\\ & = \frac{c_1}{{\varepsilon}^2 n^2},\end{aligned}\ ] ] for a non - negative constant @xmath493 since @xmath494 . hence , @xmath495 so the borel  cantelli lemma applies . in particular , almost surely @xmath496 for all but finitely many @xmath1 . furthermore , @xmath14 is not an atom of @xmath2 , so we have almost surely that , for all @xmath1 , @xmath497 where @xmath498 is an almost surely finite random variable . now , since @xmath484 , the bound in implies that , for @xmath1 sufficiently large , @xmath499 for a positive constant @xmath500 ( depending on @xmath14 ) . it follows that @xmath501 almost surely . since @xmath33 was arbitrary , the proof is complete . the reverse inequality in lemma [ lma : kabl : lnsmall ] requires an anti - concentration result that can be found , for example , in ( * ? ? ? * theorem 2.22 on page 76 ) . before stating the lemma , we define the lvy concentration function of a complex - valued random variable . suppose that @xmath506 are iid , non - degenerate , complex - valued random variables . then , there is a positive constant @xmath51 ( depending only on the distribution of @xmath507 ) , so that , for any @xmath504 , @xmath508 for all @xmath20 . [ lma : anticoncentration ]    theorem 2.22 on page 76 in @xcite implies that equation holds when @xmath509 are iid real - valued random variables and the supremum in the concentration function is taken over real numbers ( see also ( * ? ? ? * corollary 6.8 ) for a more general version of this inequality ) . we extend this to the complex case in the following way . by assumption , @xmath509 are iid and non - degenerate , so at least one of the real - valued random variables @xmath510 or @xmath511 is non - degenerate . without loss of generality , assume @xmath510 is non - degenerate . then @xmath512 the last expression is bounded by @xmath513 , for some constant @xmath51 that depends only on the distribution of @xmath510 by the previously mentioned result in @xcite . a nearly identical argument applies if @xmath510 is degenerate and @xmath511 is non - degenerate .      since @xmath66 , we assume @xmath1 is sufficiently large so that @xmath515 . fix @xmath460 , and let @xmath33 be given . since @xmath2 is non - degenerate and @xmath14 is not an atom of @xmath2 , it follows that @xmath516 are iid , non - degenerate , complex - valued random variables satisfying the hypotheses of lemma [ lma : anticoncentration ] . by absorbing the contribution of @xmath517 into the complex number @xmath518 in the definition of the concentration function , we conclude from lemma [ lma : anticoncentration ] that @xmath519 for a positive constant @xmath51 depending only on the distribution of @xmath520 . as @xmath30 , the right - hand side goes to zero ( since @xmath66 ) , which completes the proof . in order to prove lemma [ lma : kabl : intsmall ] , we will apply lemma [ lma : kabl : tauvu ] to the random functions @xmath530 , where @xmath531 is a continuous function with compact support . lemma [ lma : kabl : lnsmall ] establishes the first condition , and the tightness condition ( with @xmath532 ) follows from the next lemma . for the remainder of the paper , we let @xmath533 denote the open disk of radius @xmath534 centered about the origin . fix @xmath535 such that the support of @xmath531 is contained in the open disk @xmath536 . we will occasionally use @xmath537 to denote the indicator function of the set @xmath536 .      in view of lemma [ lma : kabl : tauvu ] , the proof of lemma [ lma : kabl : intsmall ] reduces to establishing lemma [ lma : kabl : inttight ] . we bound the integral in lemma [ lma : kabl : inttight ] by employing the poisson jensen formula as in @xcite . in order to do so , we will need a uniform bound on @xmath539 for @xmath14 of certain magnitudes , which is the content of the following lemma . the proof is similar in spirit to that of lemma [ lma : kabl : logbelowzero ] . we first claim that @xmath543 for any @xmath544 , @xmath545 , and @xmath546 . this equivalence will allow us to employ the method of lemma [ lma : kabl : logbelowzero ] and control the behavior of @xmath547 . to establish the forward direction of , observe that @xmath548 for any @xmath14 satisfying @xmath549 . hence , @xmath550 implies @xmath551 . on the other hand , if @xmath551 , write @xmath552 in polar coordinates , and note that @xmath553 has modulus @xmath554 and satisfies @xmath555 the fact that @xmath550 follows . we are ready to construct @xmath556 from two exceptional sets @xmath557 and @xmath558 . define @xmath559 it follows from the fubini tonelli theorem that @xmath557 has lebesgue measure zero since @xmath560    we now construct @xmath558 . let @xmath561 denote lebesgue measure on the real line , and let @xmath562 clearly , @xmath563 . equivalence and markov s inequality imply that for a fixed @xmath477 and @xmath478 , @xmath564 where @xmath49 is an absolute constant . it follows that @xmath565 so the borel cantelli lemma and the countability of @xmath566 show that outside of a set @xmath567 of lebesgue measure zero , @xmath568 for all but finitely many pairs @xmath486 . hence , for @xmath569 , @xmath570 where @xmath571 is a positive constant depending on @xmath572 . ( note that since @xmath573 , @xmath574 for each pair @xmath486 ) . if we define @xmath575 , then , @xmath576 has lebesgue measure zero , and for @xmath541 , we have that , for any @xmath577 and any @xmath33 , @xmath578 \\ & = \frac{c_r'}{n^2},\end{aligned}\ ] ] where we used in the first step and markov s inequality in the second . here , @xmath579 is a positive constant depending only on @xmath572 and @xmath2 . by the borel cantelli lemma , it follows that almost surely , @xmath580 for all but finitely many @xmath1 . this guarantees that for @xmath581 , there is an almost surely bounded , real - valued random variable @xmath582 for which @xmath583 almost surely . ( note that @xmath584 for all @xmath541 by the definition of the set @xmath557 . ) the last inequality holds for all sufficiently large @xmath1 by . as @xmath33 was arbitrary , now follows . we now use the poisson jensen formula to re - write @xmath585 . for any @xmath586 and @xmath477 , let @xmath587 be the roots and critical points , respectively , of @xmath0 that are located in the open disk @xmath588 . the poisson jensen formula ( see , for example , ( * ? ? ? * chapter ii.8 ) ) implies that for any @xmath589 which is not a zero or pole of @xmath457 , @xmath590 where @xmath591 and @xmath592 denotes the poisson kernel @xmath593,\ \alpha \in [ 0,2\pi ] . \label{eqn : kabl : pker}\ ] ]      fix @xmath596 . then , for any @xmath597 $ ] and @xmath598 , we have @xmath599 the last inequality follows from the fact that @xmath600 and from the equivalence @xmath601 which holds for all @xmath602 . consequently , for any @xmath596 and @xmath603 , @xmath604 therefore , we obtain @xmath605 the desired result now follows by applying lemma [ lma : kabl : unifzbound ] to . in particular , since the exceptional set @xmath540 of lemma [ lma : kabl : unifzbound ] has measure zero , we can choose @xmath594 so that holds almost surely . next , we show that @xmath606 is bounded below uniformly for @xmath596 . we assume that @xmath607 , and we first consider the case when @xmath608 . there is no loss of generality in assuming @xmath607 , for if @xmath609 , we can choose a different point @xmath610 and prove theorem [ thm : kabl : detcond ] for the random variables @xmath611 and the deterministic array @xmath612 . this follows since the translation of the roots of @xmath0 by @xmath613 simply translates the critical points by @xmath613 ( see proposition [ prop : translate ] ) . since @xmath607 , we have @xmath615 almost surely ; in other words , @xmath529 is almost surely not a pole of @xmath457 . furthermore , by lemma [ lma : kabl : logaboveneg ] , it follows that @xmath529 is not a zero of @xmath457 with probability @xmath616 . consequently , on the same event , the poisson  jensen formula applies to @xmath617 , and we obtain @xmath618 the inequality comes from eliminating @xmath619 we bound the remaining two terms in probability . a bound for the first term follows from lemma [ lma : kabl : logaboveneg ] . it remains to find a lower bound ( in probability ) for the last term in . let @xmath620 be the random and deterministic roots , respectively , of @xmath0 that are contained in @xmath588 . ( note that @xmath621 . ) the law of large numbers implies that @xmath622 almost surely as @xmath30 . the expectation on the right - hand side is finite since @xmath623 due to the assumption @xmath607 and by the bounds @xmath624 & = -{\mathbb{e}}\left[\log_{-}{\left\vert\frac{x_1}{r}\right\vert } - \log_{-}{\left\vertx_1\right\vert } + \log_{-}{\left\vertx_1\right\vert}\right]\\ & \geq-{\mathbb{e}}\left[\log_{-}\left(\frac{1}{r}\right)\right ] - { \mathbb{e}}\left[\log_{-}{\left\vertx_1\right\vert}\right ] \\ & \geq-\log(r ) - { \mathbb{e}}\left[\log_{-}{\left\vertx_1\right\vert}\right],\end{aligned}\ ] ] which follow from the fact that @xmath625 . since @xmath72 , it follows that @xmath626 \geq -\log(r ) - { \mathbb{e}}\log_{-}{\left\vertx_1\right\vert}\ ] ] almost surely as @xmath30 , and as a consequence , we have almost surely @xmath627 for some non - negative constant @xmath628 ( depending on @xmath572 ) . similarly , as @xmath625 , we have @xmath629 by condition and the fact that @xmath66 , we obtain @xmath630 ( recall that @xmath607 , and hence @xmath631 . ) together , and imply the desired conclusion . the proof presented here closely follows the arguments in @xcite . for simplicity , define @xmath633 for @xmath634 $ ] . by the definition of the poisson kernel and reasoning similar to that used to derive the bounds in , we have @xmath635 for all @xmath596 and @xmath634 $ ] . notice that @xmath636 for all @xmath637 $ ] , so we have @xmath638 it follows that , for any @xmath477 and any @xmath596 , @xmath639 in the case where @xmath640 for all @xmath634 $ ] , we obtain the bound @xmath641 otherwise , @xmath642 for all @xmath634 $ ] , and continuing from above , @xmath643 in either case , taking the infimum over all @xmath596 and applying the results of lemmas [ lma : kabl : unifzbound ] and [ lma : kabl : in0bd ] gives the desired conclusion . we complete the proof of lemma [ lma : kabl : inttight ] by applying lemma [ lma : kabl : bdpint ] and lemma [ lma : kabl : bdpintbelow ] to . let @xmath594 be as in lemma [ lma : kabl : bdpint ] . from , we apply the cauchy schwarz inequality twice to obtain @xmath644 for @xmath645 that is not a zero or pole of @xmath457 . since there are finitely many zeros and poles of @xmath457 for a fixed @xmath1 and a fixed realization of @xmath457 , implies @xmath646 almost surely . lemmas [ lma : kabl : bdpint ] and [ lma : kabl : bdpintbelow ] establish that @xmath647 for some constant @xmath49 , and hence the sequence @xmath648 is tight . the remaining two terms of are bounded almost surely . indeed , for @xmath596 and @xmath649 , we have @xmath650 and hence @xmath651 by a simple change of variables , we obtain @xmath652 and similarly @xmath653 thus , by the local integrability of the squared logarithm , @xmath654 almost surely for all @xmath655 , where @xmath656 is a constant that depends only on @xmath572 and @xmath657 , and , in the last inequality , we used the fact that @xmath658 . a similar argument applies to the integral of the sum in involving the critical points @xmath659 ; we omit the details . r. pemantle , i. rivin , _ the distribution of zeros of the derivative of a random polynomial _ , advances in combinatorics . waterloo workshop in computer algebra 2011 , i. kotsireas and e. v. zima , editors , springer , new york , 2013 . t. r. reddy , _ on critical points of random polynomials and spectrum of certain products of random matrices _ , ph.d . thesis submitted in july , 2015 at indian institute of science , bangalore . available at arxiv:1602.05298 . we now prove theorem [ thm : simple ] using theorem [ thm : main ] . indeed , let @xmath7 satisfy the assumptions of theorem [ thm : simple ] . since @xmath7 do not depend on @xmath1 , there exists @xmath122 such that , for any @xmath123 ,            [ thm : reduction ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath310 . let @xmath38 be iid random variables with distribution @xmath2 . for each @xmath20 , let @xmath67 be a triangular array of deterministic complex numbers with @xmath143 , and assume @xmath311 . fix @xmath33 , and suppose that for all sufficiently large @xmath1 , there are no values of @xmath67 in @xmath145 and there are @xmath120 values @xmath146 outside @xmath147 . then , almost surely , for @xmath1 sufficiently large , there are exactly @xmath120 critical points ( counted with multiplicity ) of the polynomial @xmath148 outside @xmath149 , and after labeling these critical points correctly , @xmath150 for each @xmath126 . the only difference between this theorem and theorem [ thm : main ] is that theorem [ thm : reduction ] assumes @xmath310 . using theorem [ thm : reduction ] , we prove theorem [ thm : main ] by applying proposition [ prop : translate ] . let @xmath2 have compact support . since @xmath103 is nonempty , choose @xmath312 . we now consider the polynomial @xmath313 where @xmath314 . let @xmath315 be the distribution of @xmath316 . then @xmath315 has compact support and @xmath317 . in addition , the sets @xmath318 and @xmath319 are translates by @xmath320 of the sets @xmath299 and @xmath103 , respectively . thus , by assumption , there are no values of @xmath321 in @xmath322 and there are @xmath120 values @xmath323 outside @xmath324 . therefore , by theorem [ thm : reduction ] and proposition [ prop : translate ] , we conclude that almost surely , for @xmath1 sufficiently large , there are exactly @xmath120 critical points of @xmath0 outside @xmath149 and after labeling correctly , @xmath325 for @xmath126 . adding @xmath326 to both sides completes the proof . [ thm : reductiondist ] let @xmath2 be a probability measure on @xmath19 with compact support , and suppose @xmath310 . let @xmath38 be iid random variables with distribution @xmath2 . for each @xmath20 , let @xmath67 be a triangular array of deterministic complex numbers with @xmath143 . fix @xmath33 , and suppose that for all sufficiently large @xmath1 , there are no values of @xmath67 in @xmath145 and there is one value @xmath153 outside @xmath147 . then , almost surely , for @xmath1 sufficiently large , there is exactly one critical point of the polynomial @xmath148 outside @xmath149 , and after labeling the critical points correctly , @xmath327    the proof of theorem [ thm : distinguished ] using theorem [ thm : reductiondist ] is nearly identical to the proof of theorem [ thm : main ] above ; we omit the details . it remains to prove theorems [ thm : reduction ] and [ thm : reductiondist ] . we prove theorems [ thm : reduction ] and [ thm : reductiondist ] simultaneously . indeed , for the first part of the proof , we continue to use the notation of theorem [ thm : reduction ] . however , the same argument applies to theorem [ thm : reductiondist ] by simply taking @xmath152 . the conclusion of the proof will require us to consider the conditions of both theorems separately . in fact , the conclusion of the proof is the only place where we require condition . for notational convenience , throughout the proof we allow the implicit constants and rates of convergence in our asymptotic notation ( such as @xmath328 ) to depend on the parameter @xmath89 without notating this dependence .    for @xmath1 sufficiently large , we decompose @xmath329 where , by assumption , @xmath330 are outside @xmath147 and @xmath331 are in @xmath94 . in addition , @xmath332 are in @xmath333 with probability @xmath334 . let @xmath180 be the diagonal matrix @xmath335 where @xmath336 and @xmath337 here , the subscripts `` @xmath338 '' and `` @xmath339 '' refer to the roots inside and outside @xmath94 , respectively . of course , @xmath180 , @xmath340 , and @xmath341 all depend on @xmath1 , but we do not denote this dependence in our notation .    by theorem [ thm : companion ] , it follows that @xmath342 , \nonumber\end{aligned}\ ] ] where @xmath219 is the identity matrix and @xmath343 is the @xmath220 all - one matrix . we decompose , @xmath344 where @xmath345 denotes the @xmath346 all - one matrix . thus , we conclude that @xmath347 we will eventually apply lemma [ lemma : block ] to compute this determinant , but first we will need to consider the upper - left block @xmath348 let @xmath349 denote the all - one @xmath1-vector ; we will often drop the subscript ( and just write @xmath350 ) when its size can be deduced from context . we will make use of the following lemma . [ lemma : technical ] under the assumptions of theorem [ thm : reduction ] ( alternatively , theorem [ thm : reductiondist ] ) , almost surely , for @xmath1 sufficiently large , the matrix @xmath351 is invertible for every @xmath352 and the function @xmath353 is analytic outside @xmath354 . in addition , almost surely @xmath355    recall that the entries of the diagonal matrix @xmath340 are contained in @xmath94 . thus , for @xmath352 , the matrix @xmath356 is invertible . in addition , since @xmath357 is a diagonal matrix , we obtain @xmath358   \nonumber \\      & = \frac{1}{n } \sum_{j=1}^{n - k_n } \frac{x_j}{z - x_j } + \frac{1}{n } \sum_{l = s+1}^{k_n } \frac{\xi_l^{(n ) } } { z - \xi_l^{(n)}}.   \label{eq : trpartial}\end{aligned}\ ] ] among other things , this implies that the function @xmath359 is analytic outside @xmath354 ; we will use this fact later to show that the function in is analytic on the same set . since @xmath103 is compact , it follows from proposition [ prop : mmu ] that @xmath94 is bounded . let @xmath360 be such that @xmath361 for all @xmath362 . let @xmath363 . then for @xmath364 , we have @xmath365 for @xmath366 and similarly @xmath367 for each @xmath368 . thus , @xmath369 in particular , this bound implies that @xmath370 for all @xmath364 . thus , we can apply lemma [ lemma : sm ] to conclude that the matrix in is invertible for every @xmath364 . indeed , since @xmath371 is at most rank one is rank one , and so the product @xmath372 is either rank one or rank zero . in fact , a simple computation reveals that the product is rank zero if and only if @xmath340 is the zero matrix . ] , it follows from lemma [ lemma : sm ] ( taking @xmath373 and @xmath374 ) that @xmath375 hence , by the bound in , we have , with probability one , @xmath376 in addition , the right - hand side of is analytic in the region @xmath364 , which implies that the function on the left - hand side is also analytic in the same region . let @xmath377 be the compact set @xmath378 . it remains to show that , almost surely , for @xmath1 sufficiently large , the matrix in is invertible for every @xmath379 , the function in is analytic in @xmath377 , and @xmath380 to establish these results we will again apply lemma [ lemma : sm ] . however , in this case , we will need more precise estimates than those established above .    indeed , returning to , we find that @xmath381 since @xmath331 are contained in @xmath94 , it follows from the triangle inequality that @xmath382 in addition , by lemma [ lemma : lln ] and the borel cantelli lemma , we have , almost surely @xmath383 as @xmath377 is compact and @xmath84 can not vanish on @xmath377 ( since @xmath384 ) , there exists @xmath95 such that @xmath385 for all @xmath379 . specifically , by the assumption that @xmath310 , it follows that @xmath386    therefore , by , , and , we conclude from that , almost surely , for @xmath1 sufficiently large , @xmath387 and @xmath388 hence , by lemma [ lemma : sm ] , we obtain for @xmath379 which , combined with the bounds above , yields @xmath389 almost surely . as before , also implies that the function in is analytic on @xmath377 . the proof of the lemma is complete . let us dispatch the simplest case of theorem [ thm : reduction ] : when @xmath390 . indeed , if @xmath390 , then @xmath391 . in this case , and the invertibility of imply that @xmath0 has no critical points outside @xmath149 , completing the proof . thus , for the remainder of the proof , we assume @xmath392 . we return to the block determinant in . by lemma [ lemma : technical ] , almost surely , for @xmath1 sufficiently large , the upper - left block is invertible for all @xmath352 . thus , by lemma [ lemma : block ] , we conclude that almost surely @xmath393 for all @xmath352 , where @xmath394 in other words , the zeros of @xmath62 outside of @xmath354 ( counted with multiplicity ) are precisely the zeros of @xmath395 outside of @xmath354 ( counted with multiplicity ) . notice that this is the determinant of an @xmath396 matrix , and @xmath397 . we have thus reduced the problem of studying an @xmath220 matrix to an @xmath396 matrix . this reduction greatly simplifies the forthcoming analysis . before we conclude the proof , we make one final observation : since @xmath398 and @xmath399 , we can rewrite the determinant in as @xmath400    we now conclude the proof of theorems [ thm : reduction ] and [ thm : reductiondist ] separately . let us begin with theorem [ thm : reduction ] . indeed , under the assumptions of theorem [ thm : reduction ] , @xmath401 ( recall that @xmath402 denotes the spectral norm of the matrix @xmath341 . ) thus , by lemma [ lemma : detdiff ] and lemma [ lemma : technical ] , we have , almost surely @xmath403 because @xmath404 . notice that the zeros of @xmath405 are precisely the values @xmath406 . in view of rouch s theorem ( since both determinants are analytic outside @xmath149 due to lemma [ lemma : technical ] ) , we conclude that , almost surely , for @xmath1 sufficiently large , @xmath0 has exactly @xmath120 critical points outside @xmath407 , and after correctly labeling the critical points , @xmath408 for each @xmath126 . this completes the proof of theorem [ thm : reduction ] . we now turn to the proof of theorem [ thm : reductiondist ] . recall that , in this case , @xmath409 . thus , the matrix in is just a @xmath410 matrix , and hence the the zeros of @xmath62 outside of @xmath149 are precisely the solutions of @xmath411 outside @xmath149 . by lemma [ lemma : technical ] , we have , almost surely , @xmath412 for some constant @xmath49 . since both these terms are analytic outside @xmath354 due to lemma [ lemma : technical ] , we can again apply rouch s theorem . however , since @xmath413 does not necessarily converge to zero , we have to be slightly more careful . let @xmath414 be any simple closed contour outside @xmath354 which satisfies @xmath415 for all @xmath416 . then , by the estimate in , rouch s theorem implies that the number of solutions to inside @xmath414 is the same as the number of zeros of @xmath417 inside @xmath414 . hence , we conclude that almost surely , for @xmath1 sufficiently large , there is exactly one critical point of @xmath0 outside @xmath354 and that critical point takes the value @xmath418 . the proof of theorem [ thm : reductiondist ] is complete .
let @xmath0 be a random , degree @xmath1 polynomial whose roots are chosen independently according to the probability measure @xmath2 on the complex plane . for a deterministic point @xmath3 lying outside the support of @xmath2 , we show that almost surely the polynomial @xmath4 has a critical point at distance @xmath5 from @xmath3 . in other words , 
 conditioning the random polynomials @xmath0 to have a root at @xmath3 , almost surely forces a critical point near @xmath3 . 
 more generally , we prove an analogous result for the critical points of @xmath6 , where @xmath7 are deterministic . 
 in addition , when @xmath8 , we show that the empirical distribution constructed from the critical points of @xmath9 converges to @xmath2 in probability as the degree tends to infinity , extending a recent result of kabluchko @xcite .