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let @xmath1 denote the symmetric group of permutations acting on a set @xmath2 of cardinality @xmath3 , and @xmath4 denote the uniform law on @xmath1 , @xmath5 a seminal concentration result on @xmath1 obtained by maurey is the following . @xcite[permmau ] let @xmath6 be the hamming distance on the symmetric group , for all @xmath7 , @xmath8 then for any subset @xmath9 such that @xmath10 , and for all @xmath11 , one has @xmath12 where @xmath13 .    milman and schechtman @xcite generalized this result to some groups whose distance is invariant by translation . for example , in the above result we may replace ( up to constants ) the hamming distance by the _ transposition distance _ @xmath14 that corresponds to the minimal number of transpositions @xmath15 such that @xmath16 . the distances @xmath17 and @xmath6 are comparable , @xmath18 ( we refer to @xcite for comments about these comparison inequalities ) .    a few years later , a stronger concentration property in terms of dependence in the parameter @xmath3 , has been settled by talagrand using the so - called `` convex - hull '' method @xcite ( see also @xcite ) . this property implies maurey s result with a slitly worse constant . let us recall some notations from @xcite . for each @xmath19 and @xmath20 , let @xmath21 be the set of vectors @xmath22 with @xmath23 for @xmath24 let @xmath25 denote the convex hull of @xmath26 in @xmath27 , @xmath28 where @xmath29 denotes the set of probability measures on @xmath30 . talagrand introduced the quantity @xmath31 with @xmath32 , that measures the distance from @xmath33 to the subset @xmath30 . [ tala95]@xcite for any subset @xmath9 , @xmath34    maurey s concentration result easily follows by observing that @xmath35 and applying tchebychev inequality with usual optimization arguments . talagrand s result has been first extended to product of symmetric groups by mcdiarmid @xcite , and then further by luczak and mcdiarmid to cover more general permutation groups which act suitably `` locally '' @xcite .    for any @xmath36 , the _ support _ of @xmath33 , denoted by @xmath37 , is the set @xmath38 and the _ degree _ of @xmath33 , denoted by @xmath39 , is the cardinality of @xmath37 , @xmath40 . by definition , according to @xcite , a subgroup @xmath0 of @xmath1 is _ @xmath41-local _ , @xmath42 , if for any @xmath43 and any @xmath44 with @xmath45 , there exists @xmath46 such that @xmath47 , @xmath48 and @xmath49 .    as explained in @xcite , any @xmath50-local group is a direct product of symmetric groups on its orbits , the alternating group ( consisting of even permutations ) is @xmath51-local , and any @xmath51-local group is a direct product of symmetric or alternating groups on its orbits .    in the present paper , the concentration result by luczak - mcdiarmid and talagrand is a consequence of a weak transport - entropy inequality satisfied by the uniform law on @xmath0 . we also prove weaker type of transport entropy inequalities for a larger class of probability measures on @xmath0 , denoted by @xmath52 .    for a better comprehension of the class of measures @xmath52 , let us first consider the case of the symmetric group @xmath1 on @xmath53:=\{1,\ldots , n\}$ ] . let @xmath54 denote the transposition in @xmath1 that exchanges the elements @xmath55 and @xmath56 in @xmath53 $ ] . one may easily check that the map @xmath57 is one to one . the set of measures @xmath52 consists of probability measures on @xmath1 which are pushed forward by the map @xmath58 of product probability measures on @xmath59 , @xmath60),\;\forall i\in [ n]\big\},\ ] ] where by definition @xmath61 for any subset @xmath62 in @xmath1 . the uniform measure @xmath4 on @xmath1 belongs to @xmath52 since @xmath63 with @xmath64 , where for each @xmath55 , @xmath65 denotes the uniform probability measure on @xmath66 $ ] . let us now construct the class of measures @xmath52 for any @xmath41-local group @xmath0 . to clarify the notations , the elements of @xmath2 are labelled with integers , @xmath67 $ ] . the _ orbit _ of an element @xmath68 , denoted by @xmath69 , is the set of @xmath70 connected to @xmath56 by the elements of @xmath0 , @xmath71 let us observe that for any @xmath72 $ ] , @xmath73 since the identity @xmath74 belongs to @xmath0 . let @xmath0 be a @xmath41-local subgroup of @xmath1 . for any @xmath75 let @xmath76 $ ] . there exists a fixed family of permutions @xmath77 with @xmath75 and @xmath78 with @xmath79 and for every @xmath80 @xmath81 , \quad \mbox{and}\quad   \deg(t_{i_jj})\leq \ell,\ ] ] such that the map @xmath82 is one to one . such a family @xmath83 is called `` @xmath41-local base of @xmath0 '' in the present paper .    as for the symmetric group , the class @xmath52 is made up of all probability measures on @xmath0 which are pushed forward of product probability measures on @xmath84 by a map @xmath85 defined by , @xmath86 where @xmath87 is a @xmath41-local base of @xmath0 . the class @xmath52 contains the uniform law on @xmath0 , obtained by choosing for each @xmath88 the uniform law on @xmath89 .    in this paper , the concentration results are derived from weak transport - entropy inequalities , involving the relative entropy @xmath90 between two probability measures @xmath91 on @xmath0 given by @xmath92 if @xmath93 is absolutely continuous with respect to @xmath4 and @xmath94 otherwise . the terminology `` weak transport - entropy '' introduced in @xcite , encompass many kinds of transport - entropy inequalities from the well - known talagrand s transport inequality satisfied by the standard gaussian measure on @xmath95 @xcite , to the usual csizr - kullback - pinsker inequality @xcite that holds for any ( reference ) probability measure @xmath4 on a polish metric space @xmath96 , namely @xmath97 where @xmath98 denotes the total variation distance between @xmath4 and @xmath93 , @xmath99 where the supremum runs over all measurable subset @xmath30 of @xmath96 . we refer to the survey @xcite for other examples of weak transport - entropy inequalities and their connections with the concentration of measure principle . the next theorem is one of the main result of this paper . it presents new weak transport inequalities for the uniform measure on @xmath0 or measures in the class @xmath52 , that recover the concentration results of theorems [ permmau ] and [ tala95 ] . we denote by @xmath6 the hamming distance on @xmath0 : for any @xmath100 , @xmath101 and the distance @xmath14 is defined as the minimal number of elements of @xmath0 , @xmath15 , with degree less than @xmath41 , such that @xmath16 . for any measures @xmath102 , the set @xmath103 denotes the set of all probability measures on @xmath104 with first marginal @xmath105 and second marginal @xmath106 . we denote by @xmath107 the wasserstein distance between @xmath105 and @xmath106 according to the distance @xmath108 or @xmath109 . @xmath110 we also consider two other optimal weak transport costs , @xmath111 and @xmath112 defined by @xmath113 and @xmath114 where @xmath115 represents a probability measure such that @xmath116 . by jensen s inequality , these weak transport costs are comparable , namely @xmath117 where the last inequality holds for @xmath108 . [ transportsymetrique ] let @xmath0 be a @xmath41-local subgroup of @xmath1 .    1 . let @xmath4 be a probability measure of the set @xmath52 defined by . then , for all probability measures @xmath105 and @xmath106 on @xmath0 , @xmath118 and @xmath119 where @xmath120 when @xmath4 is the uniform law on @xmath0 , the inequalities and hold with @xmath121 2 . let @xmath4 denotes the uniform law on @xmath0 . then , for all probability measures @xmath105 and @xmath106 on @xmath0 , @xmath122 where @xmath123 .    the proofs of these results , given in the next section , are inspired by talagrand seminal work on @xmath1 @xcite , and luczak - mcdiarmid extension to @xmath41-local groups @xcite . * comments :*    * here is a more popular dual formulation of the transport - entropy inequality : for all @xmath124-lipschitz functions @xmath125 ( with respect to the distance @xmath126 ) , @xmath127 for the uniform measure on @xmath1 , @xmath128 and this property is widely commented in @xcite ; it is also a consequence of hoeffding inequalities for bounded martingales ( see page 18 of @xcite ) . the concentration result derived from item ( a ) are of the same nature as the one obtained by the `` bounded differences approach '' in @xcite . * similarly , by proposition 4.5 and theorem 2.7 of @xcite and using the identity @xmath129 we may easily show that the weak transport - entropy inequality is equivalent to the following dual property : for any real function @xmath130 on @xmath0 and for any @xmath131 , @xmath132 where the infimum - convolution operator @xmath133 , @xmath11 , is defined by @xmath134 + moreover , let us observe that following our proof of in the next section , for each @xmath135 the inequality can be improved by replacing the square cost function by the convex cost @xmath136 given in lemma [ completegraph ] . more precisely , holds replacing @xmath137 by @xmath138 defined by @xmath139 for any @xmath140 . * proposition 4.5 and theorem 9.5 of @xcite also provide a dual formulation of the weak transport - entropy inequality : for any real function @xmath130 on @xmath0 and for any @xmath131 , @xmath141 where the infimum convolution operator @xmath142 is defined by @xmath143 as explained at the end of this section , the property directly provides the following version of the talagrand s concentration result for the uniform law on @xmath0 . + [ talgen ] let @xmath4 denotes the uniform law on a @xmath41-local subgroup @xmath0 of @xmath1 . then , for all @xmath144 and all @xmath145 , one has @xmath146 with @xmath123 . as a consequence , by tchebychev inequality , for any @xmath145 and all @xmath11 , @xmath147 + for @xmath148 , this result is exactly theorem 2.1 by luczak - mcdiarmid @xcite , that generalizes theorem [ tala95 ] on @xmath1 ( since @xmath1 is a 2-local group ) .    by projection arguments , theorem [ transportsymetrique ] applied to the uniform law @xmath4 on the symmetric group @xmath1 , also provides transport - entropy inequalities for the uniform law on the slices of the discrete cube @xmath149 . namely , for @xmath150 , let us denote by @xmath151 , @xmath152 , the slice of discrete cube defined by @xmath153 the uniform law on @xmath151 , denoted by @xmath154 , is the pushed forward of @xmath4 by the projection map @xmath155)},\end{array}\ ] ] where @xmath156):=\{\sigma(1),\ldots,\sigma(k)\}$ ] and for any subset @xmath30 of @xmath53 $ ] , @xmath157 is the vector with coordinates @xmath158 $ ] . in other terms , @xmath159 and @xmath160 for all @xmath161 . let @xmath162 denotes the hamming distance on @xmath151 defined by @xmath163    [ transportslice ] let @xmath154 be the uniform law on @xmath151 , a slice of the discrete cube .    1 .   for all probability measures @xmath105 and @xmath106 on @xmath151 , @xmath164 and @xmath165 where @xmath166 is the wasserstein distance associated to @xmath162 , @xmath167 is the weak optimal transport cost defined by with @xmath168 , and @xmath169 . 2 .   for all probability measures @xmath105 and @xmath106 on @xmath151 , @xmath170 where @xmath171 with @xmath172 for all @xmath173 .    up to constants , the weak transport inequality is the stronger one since for all @xmath174 , @xmath175 the proof of theorem [ transportslice ] is given in section [ sectionslice ] . the transport - entropy inequality is derived by projection from the transport - entropy inequality for the uniform measure @xmath4 on @xmath1 . the same projection argument could be used to reach the results of ( a ) from the transport - entropy inequality of item ( a ) of theorem [ transportsymetrique ] , but it provides worse constants . the constant @xmath176 in ( a ) is obtained by working directly on @xmath151 and following similar arguments as in the proof of theorem [ transportsymetrique ] . the results of theorem [ transportslice ] also extend to the multinomial law . let @xmath177 be a set of cardinality @xmath178 and let @xmath179 be a collection of non - zero integers satisfying @xmath180 . the multinomial law @xmath181 is by definition the uniform law on the set @xmath182 , \left|\big\{i\in[n ] , x_i = e_l\big\}\right|=k_l \bigg\}.\ ] ] for any @xmath183 , one has @xmath184 . as a result , the weak transport - entropy inequality holds on @xmath185 replacing the measure @xmath154 by the measure @xmath181 . the proof of this result is a simple generalization of the one on @xmath151 , by using the projection map @xmath186 defined by : @xmath187 if and only if @xmath188 , \,\forall i\in j_l,\ ] ] where @xmath189 , k_0+\cdots+k_{l-1 } <   i\leq k_0+\cdots+k_{l}\big\}$ ] , with @xmath190 . the details of this proof are let to the reader . a straightforward application of transport - entropy inequalities of theorem is deviation s bounds for different classes of functions . for more comprehension , we present below deviations bounds that can be reached from theorem [ transportsymetrique ] for the uniform law @xmath4 on a @xmath41-local subgroup of @xmath1 . obviously some of these results extend to the class of measures @xmath52 and a similar corollary can be derived from theorem [ transportslice ] on the slices of the discrete cube . for any @xmath191 , the mean of @xmath192 is denoted by @xmath193 . [ cordev ] let @xmath4 be the uniform law on a @xmath41-local subgroup @xmath0 of @xmath1 . let @xmath194 be a real function on @xmath0 .    1 . assume that there exists a function @xmath195 such that for all @xmath196 , @xmath197 where @xmath109 or @xmath108 . then for all @xmath198 , one has @xmath199 and @xmath200 where the constants @xmath201 and @xmath202 are defined as in item @xmath203 of theorem [ transportsymetrique ] . 2 .   assume that there exist functions @xmath204 , @xmath205 such that for all @xmath196 , @xmath206 + then , for all @xmath198 , one has @xmath207 and @xmath208 where @xmath209 and @xmath210 . * the above deviation s bounds of @xmath194 around its mean @xmath211 are directly derived from the dual representations , , of the transport - entropy inequalities of theorem [ transportsymetrique ] , when @xmath212 goes to 0 or @xmath212 goes to 1 . by classical arguments ( see @xcite ) , corollary [ cordev ] also implies deviation s bounds around a median @xmath213 of @xmath214 , but we loose in the constants with this procedure . however , starting directly from properties of type theorem [ talgen ] , one simply get the following bound under the condition of @xmath215 : for all @xmath198 , @xmath216 where @xmath217 , @xmath218 , and @xmath210 . + the idea of the proof is to choose the set @xmath219 of measure @xmath10 and to show that under the conditions of @xmath215 , @xmath220 then , the deviation bound above the median directly follows from theorem [ talgen ] by optimizing over all @xmath145 . with similar arguments , the same bound can be reached for @xmath221 . * in ( a ) , the bound above the mean is a simple consequence of . as settled in ( a ) , this bound also holds for the deviations under the mean , and it can be slitly improved by replacing @xmath222 by @xmath223 . this small improvement is a consequence of the weak transport inequality with stronger cost @xmath224 . the same kind of improvement could be reached for the deviations above the mean under additional lipschitz regularity conditions on the function @xmath225 . * let @xmath226^n\to\r$ ] be a 1-lipschitz convex function and let @xmath227^n$ ] be a fixed vector . for any @xmath43 , let @xmath228 . by applying the results of item @xmath215 ( or even ) , to the particular function @xmath229 , @xmath43 , we recover and extend to any @xmath41 subgroup @xmath0 , the deviation inequality by adamczak , chafa and wolff @xcite ( theorem 3.1 ) obtained from theorem [ tala95 ] by talagrand . namely , since for any @xmath100 , @xmath230 with @xmath231 , corollary [ cordev ] implies , for any choice of vector @xmath227^n$ ] , @xmath232 this concentration property on @xmath1 ( @xmath233 ) plays a key role in the approach by adamczak and al . @xcite , to study the convergence of the empirical spectral measure of random matrices with exchangeable entries , when the size of the matrices is increasing . * the aim of this paper is to clarify the links between talagrand s type of concentration results on the symmetric group and functional inequalities derived from the transport - entropy inequalities . for brevity s sake , applications of these functional inequalities will not be examined in the present paper . however , let us briefly mentioned some other applications using concentration results on the symmetric group : the stochastic travelling salesman problem for sampling without replacement ( see appendix @xcite ) , graph coloring problems @xcite . we also refer to the surveys and books @xcite for other numerous examples of application of the concentration of measure principle in randomized algorithms . we start with the proof of ( b ) . under the conditions on the function @xmath194 , for any @xmath234 @xmath235 let @xmath236 . plugging this estimate into the definition of @xmath237 , we get for any @xmath43 @xmath238 as @xmath212 goes to 1 , applied to the function @xmath239 yields @xmath240 and therefore @xmath241 as @xmath212 goes to 0 , yields @xmath242 and therefore @xmath243 the deviation bounds of ( b ) then follows from and by tchebychev inequality by optimizing over all @xmath236 .    the deviation bounds of ( a ) are also obtained from by tchebychev inequality . as above , the improvement for the deviation under the mean is a consequence of applied to @xmath239 , as @xmath212 goes to 0 , and using the estimate @xmath244    take a subset @xmath245 and consider the function @xmath246 which takes the values @xmath247 on @xmath30 and @xmath248 on @xmath249 . it holds @xmath250 } \ { \lambda ( 1-\beta ) + \psi(\beta,\sigma)\},\end{aligned}\ ] ] denoting by @xmath251 so it holds @xmath252 } \ { \lambda ( 1-\beta ) + \psi(\beta,\sigma)\ } ;    \inf_{\beta \in [ 1-\varepsilon,1 ] } \ { \lambda ( 1-\beta ) + \psi(\beta,\sigma)\},\right ) \\ & \geq   \min\left ( \lambda \varepsilon ; \inf_{\beta \geq 1-\varepsilon } \psi(\beta,\sigma )        \right ) \to   \inf_{\beta \geq 1-\varepsilon } \psi(\beta,\sigma),\end{aligned}\ ] ] as @xmath253 it is easy to check that for any fixed @xmath33 , the function @xmath254 is continuous on @xmath255 $ ] , so letting @xmath256 go to @xmath247 , we get @xmath257 on the other hand , @xmath258 for all @xmath248 . this proves that @xmath259 . applying to @xmath246 and letting @xmath260 go to infinity yields to @xmath261 it remains to observe that @xmath262 . let @xmath0 be a @xmath41-local subgroup of @xmath1 and @xmath263 be a @xmath41-local base of @xmath0 . let @xmath4 be a probability measure in the set @xmath52 . by definition of @xmath52 , there exists a product probability measure @xmath264 such that @xmath265 where the map @xmath266 is defined by . each transport - entropy inequality of theorem [ transportsymetrique ] is obtained by induction over @xmath3 and using the following partition @xmath267 of the group @xmath0 : for any @xmath268 , @xmath269 if @xmath270 , then @xmath271 and if @xmath272 , then @xmath273 . in any case , from the definition of the @xmath41-local property , we may easily check that @xmath274 is a @xmath41-local group that realizes a natural embedding of a @xmath41-local subgroup of @xmath275 and @xmath276 is a @xmath41-local base of this subgroup . moreover , it is rather easy to get that for any @xmath268 , @xmath277 is the coset defined by @xmath278 . from the definition of @xmath4 , if @xmath279 , then there exist @xmath280 such that @xmath281 and therefore @xmath282 as a consequence , one has @xmath283 . let @xmath284 denote the restriction of @xmath4 to @xmath277 defined by @xmath285 from the construction of @xmath4 , @xmath286 . moreover , for all @xmath287 , @xmath288 , one has @xmath289 this property is needed in the induction step of the proofs . let us note that if @xmath55 and @xmath290 are elements of @xmath291 , then from the @xmath41-local property of @xmath0 , there exists @xmath292 such that @xmath293 and @xmath294 . we also have @xmath295 . if moreover the measure @xmath4 is the uniform law on @xmath0 , then for any @xmath296 , @xmath297 . in that case we will use in the proofs the following property : for any @xmath298 , @xmath299 and @xmath300    since the distance @xmath166 satisfies a triangular inequality , the transport - entropy inequality follows from the following one : for all probability measures @xmath105 on @xmath0 , @xmath301 a dual formulation of this property given by theorem 2.7 in @xcite and proposition 3.1 in @xcite is the following : for all functions @xmath130 on @xmath0 and all @xmath236 , @xmath302 with @xmath303 we will prove the inequality by induction on @xmath3 . when @xmath304 , either @xmath305 and the inequality is obvious , either @xmath306 is the two points space , @xmath233 and one has @xmath307 in that case , exactly corresponds to the following dual form of the csiszar - kullback - pinsker inequality ( see proposition 3.1 in @xcite ): for any probability measure @xmath93 on a polish space @xmath308 , for any measurable function @xmath309 , @xmath310 with @xmath311    the recurrence step will be also a consequence of . let @xmath312 be the partition of @xmath0 defined by . any @xmath234 admits a unique decomposition defined by @xmath313 thus , we define a probability measure @xmath314 on @xmath291 . in particular , according to the definition of the measure @xmath315 , one has @xmath316 it follows that @xmath317 where the last equality is a consequence of property . now , we will bound the right - hand side of this equality by using the induction hypotheses .    for any function @xmath318 and any @xmath319 , let @xmath320 denote the function defined by @xmath321 . for any function @xmath322 and any @xmath298 , let us note @xmath323 the next step of the proof relies on the following lemma . [ gateau ] let @xmath324 , for any function @xmath325 and any @xmath298 , one has    1 . @xmath326 + where @xmath327 if @xmath108 and @xmath328 if @xmath109 . 2 .   @xmath329 + where @xmath330 if @xmath108 and @xmath331 if @xmath109 , and @xmath332 denotes an element of @xmath0 with @xmath294 and such that @xmath293 . this lemma is obtained using the decomposition of the measures @xmath234 on the @xmath333 s . let @xmath298 . by the triangular inequality and using the invariance by translation of the distance @xmath126 , one has @xmath334 and therefore , since @xmath335 with @xmath336 if @xmath108 and @xmath328 if @xmath109 , @xmath337 it follows that @xmath338 \hat p(l)+c(\ell)\sum_{l\in \orb(n ) } \1_{l\neq i } \hat p(l ) \right\}\\ & = \inf_{\hat p\in \p({\orb(n)})}\inf_{q_l\in \p(h_n),l\in\orb(n)}\\ & \qquad\qquad \left\ { \sum_{l\in \orb(n)}\left [ \int \varphi^{t_{l , n } } \,dq_l+ \int_{h_n } d(\sigma , \tau ) dq_l(\tau)\right ] \hat p(l)+c(\ell)\sum_{l\in \orb(n ) } \1_{l\neq i } \hat p(l ) \right\}\\ & = \inf_{\hat p\in \p(\orb(n))}\left\{\sum_{l\in \orb(n ) } q^{h_n } \varphi^{t_{l , n}}(\sigma ) \hat p(l ) + c(\ell)\sum_{l\in \orb(n ) } \1_{l\neq i } \hat p(l)\right\}.\end{aligned}\ ] ] the proof of the second inequality of lemma [ gateau ] is similar , starting from the following triangular inequality @xmath339 with @xmath340 if @xmath108 and @xmath331 if @xmath109 . the end of the proof of the second inequality of lemma [ gateau ] is let to the reader .    the induction step of the proof of continues by applying consecutively lemma [ gateau ] , the hlder inequality , and the induction hypotheses to the measure @xmath341 . if @xmath342 then @xmath343 and @xmath344    if @xmath345 then @xmath346 and for any @xmath268 , @xmath347\\ & = \exp \left[\lambda\inf_{\hat p\in \p(\orb(n))}\left\{\sum_{l\in\orb(n)}\hat \varphi(l ) \hat p(l)+c(\ell )   \sum_{l\in\orb(n ) } \1_{l\neq i } \hat p(l)\right\}+k_{n-1}c(\ell)^2\frac{\lambda^2}8 \right],\end{aligned}\ ] ] where , by using property , @xmath348 . let us consider again the above infimum - convolution @xmath349 defined on the space @xmath350 , with @xmath351 , one has @xmath352 by applying with the probability measure @xmath353 on @xmath291 , the previous inequality gives @xmath354=\exp\left[\lambda \int \varphi \,d\mu + k_{n}c(\ell)^2\frac { \lambda^2}8\right].\end{aligned}\ ] ]    the scheme of the induction proof of , with a better constant @xmath201 when @xmath4 is the uniform measure on @xmath0 , is identical , starting from the second result of lemma [ gateau ] and using the property . this is let to the reader . we now turn to the induction proof of the dual formulation of the weak transport - entropy inequality . the sketch of the proof is identical to the one of .    for the initial step @xmath304 , in the non - trivial case @xmath306 and @xmath233 , and one may easily check that @xmath355 in that case , the result follows from the following infimum - convolution property . [ completegraph ] for any probability measure @xmath93 on a polish metric space @xmath308 , for all @xmath135 and all measurable functions @xmath309 , bounded from below @xmath356 where for all @xmath357 , @xmath358 and @xmath359 is the convex function defined by @xmath360 observing that @xmath361 for all @xmath362 $ ] , the above inequality also holds replacing @xmath363 by @xmath364    the proof of this lemma can be founded in @xcite ( inequality ( 4 ) ) . for a sake of completeness , we give in appendix a new proof of this result on finite spaces @xmath96 by using a localization argument ( lemma [ chose ] ) . let us now present the key lemma for the induction step in the proof . for any function @xmath322 and any @xmath298 , we define @xmath365 here , writing @xmath366 , we omit the dependence in @xmath201 to simplify the notations . the proof relies on the following lemma . [ lemmeqtilde ] let @xmath268 . for any function @xmath325 and any @xmath298 , one has    1 . @xmath367 with @xmath327 if @xmath108 and @xmath328 if @xmath109 . 2 .   @xmath368 where @xmath330 if @xmath108 and @xmath331 if @xmath109 . the proof of this lemma is similar to the one of lemma [ gateau ] . by and the inequality @xmath369 we get for any @xmath370 , @xmath371 it follows that for any @xmath298 , @xmath372 \hat p(l)+\frac{1}{2(1-s)k_n}\bigg(\sum_{l\in\orb(n ) } \1_{l\neq i }    \hat p(l)\bigg)^2 \right\}\\ & = \inf_{\hat p\in \p(\orb(n))}\inf_{q_l\in \p(h_n),l\in\orb(n)}\\ & \quad \left\ { \sum_{l\in\orb(n)}\left [ \int \varphi^{t_{l , n } } \,dq_l+\frac{1}{2c(\ell)^2sk_n } \left(\int_{h_n } d(\sigma,\tau ) dq_l(\tau ) \right)^2\right ] \hat p(l)+\frac{1}{2(1-s)k_n}\bigg(\sum_{l\in\orb(n ) } \1_{l\neq i }    \hat p(l)\bigg)^2 \right\}\\ & = \inf_{\hat p\in \p(\orb(n))}\left\{\sum_{l\in\orb(n ) } \widetilde q^{h_n}_{k_{n-1 } } \varphi^{t_{l , n}}(\sigma ) \hat p(l ) + \frac 12 \bigg(\sum_{l\in\orb(n ) } \1_{l\neq i }    \hat p(l)\bigg)^2\right\},\end{aligned}\ ] ] where the last equality follows by choosing @xmath373 , which ends the proof of the first inequality of lemma [ lemmeqtilde ] . the second inequality of lemma [ lemmeqtilde ] is obtained identically starting from . we now turn to the induction step of the proof . by the decomposition of the measure @xmath4 on the @xmath277 s , we want to bound @xmath374 where the last equality is a consequence of property .    if @xmath271 , then the result simply follows for the induction hypotheses apply to the measure @xmath341 .    if @xmath375 , then applying lemma [ lemmeqtilde ] , the hlder inequality , and the induction hypotheses , we get @xmath376\right\}\\ & \leq   \inf_{\hat p\in \p(\orb(n ) ) } \left\ { \prod_{l\in\orb(n)}\left(\int e^{-(1-\alpha )    \varphi^{t_{l , n } } } d\mu_n \right)^{-\frac{\hat p(l)\alpha}{1-\alpha}}\exp\left[\frac 12 \bigg(\sum_{l\in\orb(n ) } \1_{l\neq i }    \hat p(l)\bigg)^2\right]\right\}\\ & = \exp \left[\alpha \inf_{\hat p\in \p(\orb(n))}\left\{\sum_{l\in\orb(n ) } \hat \varphi(l ) \hat p(l)+\frac 12 \bigg(\sum_{l\in\orb(n ) } \1_{l\neq i }    \hat p(l)\bigg)^2 \right\ } \right],\end{aligned}\ ] ] where by property , @xmath377 according to the definition of the infimum convolution @xmath378 on the space @xmath379 given in lemma [ completegraph ] , the last inequality is @xmath380 and therefore lemma [ completegraph ] provides @xmath381 the proof of is completed for any measure @xmath315 . to improve the constant when @xmath4 is the uniform probability measure on @xmath0 , the proof is similar using the second inequality of lemma [ lemmeqtilde ] together with property . we prove the dual equivalent property as a consequence of the stronger following result , for any real function @xmath130 on @xmath0 , for any @xmath382 @xmath383 with where the infimum convolution operator @xmath384 is defined by , for @xmath43 @xmath385\setminus \{j\ } } \left(\int \1_{\sigma(k)\neq y(k ) } dp(y)\right)^2 \right\}\label{defqj}.   \end{gathered}\ ] ]    the proof of relies on lemma [ completegraph ] and the following . for this proof , for any @xmath43 , we define @xmath386 and for @xmath387 $ ] , @xmath388 \setminus \{j\ } } \left(\int \1_{\sigma(k)\neq y(k ) } dp(y)\right)^2 \right\ } . \end{aligned}\ ] ]    [ inverse ] let @xmath389 $ ] . for any @xmath43 , one has @xmath390 where @xmath391 . this result follows from the change of variables @xmath392 in the definition of @xmath393 , one has @xmath394 where for the last equality , we use the fact that the map that associates to any measure @xmath234 the image measure @xmath395 under the application @xmath396 , is one to one from @xmath397 to @xmath398 . here is the key lemma for the induction step of the proof of . [ lemclef ]    1 . let @xmath399 . for any @xmath287 , one has @xmath400 2 . if @xmath401 , let @xmath402 with @xmath403 , and let @xmath404 such that @xmath405 and @xmath406 . we note @xmath407 and @xmath408 . for any @xmath287 , for any @xmath409 $ ] one has @xmath410+\frac12(1-\theta)^2.\end{aligned}\ ] ]    the first part of this lemma follows from the fact that @xmath411 and the fact that @xmath412 for @xmath298 and @xmath413 . therefore , according to the definition of @xmath414 , one has for @xmath415 , @xmath416\setminus \{j\}}\left(\int \1_{\sigma   t_{i , n}(k)\neq y(k ) } dp(y)\right)^2 \right\}\\    & = \inf_{q\in { \mathcal p}(h_n ) } \left\ {   \int \varphi^{t_{j , n } } dq + \frac 1{2c(\ell)^2}\sum_{k\in [ n]\setminus \{j\}}\left(\int \1_{\sigma t_{j , n}(k)\neq yt_{i , n}(k ) } dq(y)\right)^2 \right\ }     = q^{h_n}\varphi ^{t_{j , n}}(\sigma ) . \end{aligned}\ ] ]    for the proof of the second part of lemma [ lemclef ] , let us consider @xmath417 , a collection of measures in @xmath418 , and @xmath419 ( @xmath420 ) . for @xmath421 $ ] , @xmath422,\ ] ] is a probability measure on @xmath0 . according to the definition of @xmath423 , for any @xmath298 , @xmath424 + \frac 1{2c(\ell)^2}(a+b+c )   , \ ] ] with @xmath425\setminus \supp(t_{i , j } ) } \left(\int \1_{\sigma t_{i , n}(k)\neq y(k ) } dp(y)\right)^2 , \quad b=\sum_{k\in d_i } \left(\int \1_{\sigma t_{i , n}(k)\neq y(k ) } dp(y)\right)^2,\ ] ] and @xmath426 .    since @xmath298 and @xmath427 , one has @xmath428 and @xmath429 . it follows that @xmath430 for any @xmath431 $ ] and @xmath432 , let us note @xmath433 by the cauchy - schwarz inequality , one has @xmath434\setminus \supp(t_{i , j } ) } u_i^2(k , l ) + ( 1-\theta ) \sum_{k\in [ n]\setminus \supp(t_{i , j } ) } u_j^2(k)\right].\ ] ] we also have @xmath435\\ & \leq \sum_{k\in d_i } \left [ \frac { 2\theta^2 } d u_i^2(k , k)+2(1-\theta)^2 + \frac{\theta^2 }   d\sum_{l\in d_i\setminus \{k\}}u_i^2(k , l)\right]\\ & \leq   2d^2(1-\theta)^2 + \frac \theta d \sum_{l\in d_i }   \left [ 2u_i^2(l , l ) + \sum_{k\in d_i\setminus \{l\ } } u_i^2(k , l)\right]\end{aligned}\ ] ] all these estimates together provide @xmath436\setminus \{i , l\ } } u_i^2(k , l)\right ) + ( 1-\theta ) \sum_{k\in [ n]\setminus \supp(t_{i , j } ) } u_j^2(k)\right ] . \end{gathered}\ ] ] since @xmath437 , we get by optimizing over all @xmath438 and all @xmath439 , @xmath440 + \frac 12 ( 1-\theta)^2   , \ ] ] with @xmath441\setminus \{i , l\ } } \int\1_{\sigma t_{i , n}(k)\neq y(k ) } dp_i(y)\right\}\\ & = \inf_{q_i\in\p(h_n)}\left\{\int \varphi^{t_{i , n } } dq_i+\frac 1 { c(\ell)^2}\int \1_{\sigma ( t_{i , n}(l))\neq y(t_{i , n}(l ) ) } dq_i(y ) + \frac1 { 2c(\ell)^2}\sum_{k\in [ n-1]\setminus \{t_{i , n}(l)\ } } \int\1_{\sigma t_{i , n}(k)\neq y(k ) } dp_i(y)\right\}\\ & = q^{h_n , t_{i , n}(l ) } \varphi^{t_{i , n}}(\sigma ) \end{aligned}\ ] ] and @xmath442\setminus \supp(t_{i , j } ) } \int \1_{\sigma t_{i , n}(k)\neq y(k ) } dp_i(y)\right\}\\ & = \inf_{p_j\in\p(h_j)}\left\{\int \varphi dp_j+\frac 1{2 c(\ell)^2 } \sum_{k\in [ n]\setminus \supp(t_{i , j } ) } \int\1_{\sigma t_{i , n}(k)\neq y t_{i , j}(k ) } dp_j(y)\right\}\\ & \leq \inf_{p_j\in\p(h_j)}\left\{\int \varphi dp_j+\frac 1{2 c(\ell)^2 } \sum_{k\in [ n]\setminus \{i\ } } \int\1_{\sigma t_{i , n}(k)\neq y t_{i , j}(k ) } dp_j(y)\right\}\\ & \leq \inf_{q_j\in\p(h_n)}\left\{\int \varphi^{t_{i , n}t_{i , j}^{-1 } } dq_j+\frac 1{2 c(\ell)^2 } \sum_{k\in [ n]\setminus \{i\ } } \int\1_{\sigma t_{i , n}(k)\neq y t_{i , n}(k ) } dp_j(y)\right\}\\ & = \inf_{q_j\in\p(h_n)}\left\{\int \varphi^{t_{i , n}t_{i , j}^{-1 } } dq_j+\frac 1{2 c(\ell)^2 } \sum_{k\in [ n-1 ] } \int\1_{\sigma ( k)\neq y ( k ) } dp_j(y)\right\}\\ & = q^{h_n}\varphi^{t_{i , n}t_{i , j}^{-1}}\end{aligned}\ ] ] where we use successively the following arguments : @xmath443 for any @xmath431\setminus \supp(t_{i , j})$ ] , the set @xmath53\setminus \supp(t_{i , j})$ ] is a subset of @xmath53\setminus \{i\}$ ] , for any @xmath444 one has @xmath445 . this ends the proof of lemma [ lemclef ] . we will know prove by induction over @xmath3 . for @xmath304 , in the non - trivial case , the set @xmath0 is the @xmath50 points space @xmath446 which is @xmath50-local . for @xmath447 , and for any @xmath448 , @xmath449 as a consequence , we get the expected result from lemma [ completegraph ] applied with @xmath450 . let us now present the induction step . we assume that holds at the rank @xmath451 for all @xmath452 . by symmetry , we may assume without loss of generality that @xmath453 in since @xmath4 is the uniform probability measure on @xmath0 . we assume that @xmath401 , otherwise the induction step is obvious . we first apply lemma [ inverse ] , @xmath454 let @xmath455 . according to the decomposition of the measure @xmath4 on the sets @xmath456 , @xmath457 where @xmath284 is the uniform probability measure on @xmath277 . for @xmath458 , let us note @xmath459 we choose @xmath460 such that @xmath461    by property and then applying lemma [ lemclef ] , we get @xmath462 by the induction hypotheses applied to the measure @xmath341 , it follows that @xmath463 let us now consider @xmath403 , @xmath324 , property , the second part of lemma [ lemclef ] and jensen s inequality yield : for any @xmath409 $ ] , @xmath464+\frac{\alpha}2(1-\theta)^2\right\}\\\end{aligned}\ ] ] by the induction hypotheses applied with the uniform measure @xmath341 on the @xmath41-local subgroup @xmath274 , and from property , it follows that @xmath465 according to the definition of the infimum - convolution operator @xmath466 defined on the space @xmath467 , we may easily check that for every @xmath268 , @xmath468 }   \left\{\theta \hat g ( i ) + ( 1-\theta)\min_{k\in\orb(n ) } \hat g ( k)+\frac12 ( 1-\theta)^2 \right\}.\ ] ] therefore optimizing over all @xmath421 $ ] , we get from and : for all @xmath268 , @xmath469 finally , from lemma [ completegraph ] applied with the uniform probability measure @xmath470 on @xmath291 , the equality gives @xmath471 the proof of is completed . we adapt to the space @xmath151 the proof of item ( a ) of theorem[transportsymetrique ] . in order to avoid redundancy , we only present the main steps of the proof .    by duality , it sufises to prove that for all functions @xmath130 on @xmath151 and all @xmath236 , @xmath472 where @xmath473 and for any @xmath131 , @xmath474 where for @xmath475 , @xmath476    the proof is by induction over @xmath3 and @xmath477 .    for any @xmath150 , if @xmath478 or @xmath479 , the set @xmath151 is reduced to a singleton and the inequalities or are obvious .    for @xmath304 and @xmath480 , @xmath151 is a two points set , and directly follows from property and lemma [ completegraph ] on @xmath481 . for the induction step , we consider the collection of subset @xmath482 , with @xmath483 , defined by @xmath484 since for any @xmath485 , @xmath486 any probability measure @xmath487 on @xmath151 admits a unique decomposition defined by @xmath488 thus , we define probability measures @xmath489 and a probability measure @xmath314 on the set @xmath490 . for the uniform law @xmath4 on @xmath151 , one has @xmath491 where @xmath492 is the uniform law on @xmath493 , @xmath494 , for any @xmath495 .    for any @xmath496 , let @xmath497 denote the map that exchanges the coordinates @xmath498 by @xmath499 and @xmath500 by @xmath501 for any point @xmath161 . this map is one to one from @xmath493 to @xmath502 . for any @xmath503 , the set @xmath504 can be identify to @xmath505 and therefore the induction hypotheses apply for the uniform law @xmath492 on @xmath493 with hamming distance @xmath506\setminus\{i , j\ } } \1_{x_k\neq y_k},\qquad x , y\in \omega^{i , j}.\ ] ]    for any function @xmath507 and any @xmath495 , we define @xmath508 and @xmath509    the key lemma of the proof that replaces lemma [ gateau ] and [ lemmeqtilde ] is the following . [ lemslice ] for any function @xmath510 and any @xmath495 , one has @xmath511 and @xmath512    the proof of this lemma is obtained by decomposition of the measures @xmath513 on the sets @xmath493 , and using the following inequality@xmath514 for any @xmath495 , @xmath515 . finally , the proof of the induction step by using lemma [ lemslice ] and the identity @xmath516 is let to the reader . we will explain the projection argument on the dual formulations of the transport - entropy inequalities . according to proposition 4.5 and theorem 9.5 of @xcite , the weak transport - entropy inequality is equivalent to the following property that we want to establish : for any real function @xmath517 on @xmath151 and for any @xmath131 , @xmath518 where @xmath519    let us apply property to the function @xmath520 . since @xmath159 , we get @xmath521 the inequality is an easy consequence of the following result . [ tartine ] for any @xmath36 , @xmath522    it remains to prove this lemma . by definition , one has @xmath523\right\}.\end{aligned}\ ] ] let @xmath524 such that @xmath525 . @xmath526 for @xmath527 , let us note @xmath528 , y_i=1\}$ ] . then @xmath529 if and only if @xmath530)=y$ ] . assume that @xmath531 $ ] , if @xmath530)=y$ ] and @xmath532 then @xmath533 . therefore one has @xmath534)=y , \sigma(j)\not\in y \big\}\subset \big\{\tau , u(\tau)=y,\sigma(j)\neq \tau(j)\big\}.\ ] ] assume now that @xmath535 $ ] , if @xmath530)=y$ ] and @xmath536 then we also have @xmath533 . it follows that @xmath534)=y , \sigma(j)\in y \big\}\subset \big\{\tau , u(\tau)=y,\sigma(j)\neq \tau(j)\big\}.\ ] ] from these observations , we get @xmath537 }   \left(\int \1_{u(\tau)=y,\sigma(j)\not\in y } dp(\tau)\right)^2 + \sum_{j\in [ n]\setminus[k ] }   \left(\int \1_{u(\tau)=y,\sigma(j)\in y } dp(\tau)\right)^2\\ & = \sum_{j\in [ k ] }   \left(\int \1_{\sigma(j)\not\in y } dq(y)\right)^2 + \sum_{j\in [ n]\setminus[k ] }   \left(\int \1_{\sigma(j)\in y } dq(y)\right)^2\\ & = \sum_{i\in \sigma([k ] ) }   \left(\int \1_{i\not\in y } dq(y)\right)^2 + \sum_{i\not\in \sigma([k ] ) }   \left(\int \1_{i\in y } dq(y)\right)^2\\ & = \sum_{i\in \sigma([k ] ) }   \left(\int \1_{y_i=0 } dq(y)\right)^2 + \sum_{i\not\in \sigma([k ] ) }   \left(\int \1_{y_i=1 } dq(y)\right)^2\\\end{aligned}\ ] ] setting @xmath538 , it follows that @xmath539\\ & = \sum_{i=1}^n \left(\int \1_{y_i\neq x_i } dq(y)\right)^2.\end{aligned}\ ] ] this inequality provides @xmath540 the proof of lemma [ tartine ] and item ( b ) of theorem [ transportslice ] is completed . let @xmath135 and @xmath517 be a real function on the finite set @xmath96 . we want to show that for any probability measure @xmath93 on @xmath96 , @xmath541 we will apply the following lemma whose proof is given at the end of this section . [ chose ] let @xmath542 be a real function on @xmath96 and @xmath543 . let us consider the set @xmath544 if @xmath545 is not empty , then the extremal points of this convex set are dirac measures or convex combinations of two dirac measures on @xmath96 .    given a real function @xmath517 on @xmath96 , for any @xmath543 , let @xmath546 one has @xmath547 the supremum of the linear function @xmath548 on the non empty convex set @xmath549 is reached at an extremal point of @xmath549 . therefore , by lemma [ chose ] , we get @xmath550 } \left ( ( 1-\lambda)e^ { \alpha \widetilde r^\alpha f(x)}+\lambda e^ { \alpha \widetilde r^\alpha f(y ) } \right)^{1/\alpha}\left((1-\lambda ) e^{-(1-\alpha)f(x ) } + \lambda e^{-(1-\alpha)f(y)}\right)^{1/(1-\alpha)}\end{gathered}\ ] ] now , let @xmath551 and @xmath552 be some fixed points of @xmath96 . it remains to show that for any real function @xmath517 on @xmath553 and for any @xmath554 , @xmath555    the left - hand side of this inequality is invariant by translation of the function @xmath517 by a constant . therefore , by symmetry , we may assume that @xmath556 . it follows that @xmath557 . therefore we want to check that for any non - negative function @xmath517 on @xmath558 , for any @xmath559 $ ] , @xmath560 or equivalently , setting @xmath561 , @xmath562 since @xmath563 is a convex function on @xmath255 $ ] . so , it suffices to check that @xmath564 , where @xmath565 the function @xmath566 is concave and @xmath567 . for all @xmath218 , one has @xmath568 the function @xmath569 is a bijection from @xmath570 to @xmath571 $ ] . it follows that @xmath572 } \left\{\theta h + c_\alpha ( 1-\theta)\right\},\quad h\geq 0,\ ] ] where @xmath359 is the convex function defined by @xmath573.\ ] ] after computations , we get @xmath574 and therefore we exactly have for any @xmath357 , @xmath575 } \left\{\theta f(x ) + c_\alpha ( 1-\theta)\right\}=\widetilde r^\alpha f(x).\ ] ] the proof of lemma [ completegraph ] is completed .    we will show that , if @xmath576 is a convex combination of three probability measures @xmath577 , @xmath578 with @xmath579 , @xmath580 , @xmath581 , and @xmath582 , and @xmath583 , @xmath584 , @xmath585 , then there exists two measures @xmath586 in @xmath545 and @xmath587 $ ] such that @xmath588      we will assume that @xmath592 . the case @xmath593 can be treated identically and the proof in that case is let to the reader . since @xmath592 and @xmath594 , there exists @xmath595 $ ] such that @xmath596 so , we may now assume that @xmath604 and @xmath605 and therefore @xmath606 . in that case , we exactly have @xmath607 let us choose @xmath608 the equalities ensure that @xmath609 and @xmath610 . the proof of lemma [ chose ] ends by checking that @xmath611 . one has @xmath612 according to the definitions of @xmath613 , we may easily check that @xmath614 , and @xmath615 since @xmath616 , one has @xmath617 and therefore @xmath618 as a consequence @xmath619 and according to , we get @xmath620                          c.  mcdiarmid . on the method of bounded differences . in _ surveys in combinatorics , 1989 ( norwich , 1989 ) _ , volume 141 of _ london math . lecture note ser . _ , pages 148188 . cambridge univ . press , cambridge , 1989 .
following talagrand s concentration results for permutations picked uniformly at random from a symmetric group @xcite , luczak and mcdiarmid have generalized it to more general groups @xmath0 of permutations which act suitably ` locally ' . 
 here we extend their results by setting transport - entropy inequalities on these permutations groups . 
 talagrand and luczak - mc - diarmid concentration properties are consequences of these transport - entropy inequalities . 
 we also consider transport - entropy inequalities on @xmath0 for a larger class of measures . by projection 
 , we derive transport - entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law . 
 the transport - entropy inequalities settled in this paper are new examples , in discrete setting , of weak transport - entropy inequalities introduced in @xcite .