the spectral radius of large random matrices let m ij i 1 2 ldots j 1 2 ldots be iid random variables with em 11 0 and em 2 11 sigma 2 for each n define m n m ij 1 leq i j leq n the n times n matrix whose i j component is m ij we show that lim sup n rightarrow infty rho n leq sigma a s where rho n is the spectral radius of m n sqrt n evidence from computer experiments indicates that in fact rho n rightarrow sigma a s

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

the monge ampere equation and its link to optimal transportation we survey old and new regularity theory for the monge ampere equation show its connection to optimal transportation and describe the regularity properties of a general class of monge ampere type equations arising in that context

multi to one dimensional optimal transport we consider the monge kantorovich problem of transporting a probability density on m to another on the line so as to optimize a given cost function we introduce a nestedness criterion relating the cost to the densities under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time this map is continuous if the target density has connected support we use level set dynamics to develop and quantify a local regularity theory for this map and the kantorovich potentials solving the dual linear program we identify obstructions to global regularity through examples more specifically fix probability densities f and g on open sets x m and y n with m n 1 consider transporting f onto g so as to minimize the cost s x y we give a nondegeneracy condition on s c1 1 that ensures the set of x paired with g a e y y lie in a codimension n submanifold of x specializing to the case m n 1 we discover a nestedness criterion relating s to f g that allows us to construct a unique optimal solution in the form of a map f x y when s c2 w3 1 and g and f are bounded the kantorovich dual potentials u satisfy cloc1 1 y and the normal velocity v of f 1 y with respect to changes in y is given by v x syy dxsy x f x positivity of v locally implies a lipschitz bound on f moreover c2 if f 1 y intersects x c1 transversally on subsets where this nondegeneracy positivity and transversality can be quantified for each integer r 1 the norms of u cr 1 1 and f cr 1 are controlled by these bounds g log f x cr 1 1 x w2 1 s cr 1 1 and the smallness of f 1 y we give examples showing regularity extends from x to part of x but not from y to y we also show that when s remains nested for all f g the problem in m reduces to a supermodular problem in 2017 wiley periodicals inc

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

circular law and arc law for truncation of random unitary matrix let v be the m m upper left corner of an n n haar invariant unitary matrix let 1 m be the eigenvalues of v we prove that the empirical distribution of a normalization of 1 m goes to the circular law that is the uniform distribution on z c z 1 as m with m n 0 we also prove that the empirical distribution of 1 m goes to the arc law that is the uniform distribution on z c z 1 as m n 1 these explain two observations by yczkowski and sommers 2000

how many entries of a typical orthogonal matrix can be approximated by independent normals we solve an open problem of diaconis that asks what are the largest orders of p n and q n such that z n the p n x q n upper left block of a random matrix r n which is uniformly distributed on the orthogonal group o n can be approximated by independent standard normals this problem is solved by two different approximation methods first we show that the variation distance between the joint distribution of entries of z n and that of p n q n independent standard normals goes to zero provided p n o n and q n o n we also show that the above variation distance does not go to zero if p n x n and q n y n for any positive numbers x and y this says that the largest orders of p n and q n are o n 1 2 in the sense of the above approximation second suppose n ij n n is generated by performing the gram schmidt algorithm on the columns of y n y ij n n where y ij 1 i j n are i i d standard normals we show that e n m max 1 i n 1 j m n ij y ij goes to zero in probability as long as m m n o n log n we also prove that e n m n 2 in probability when m n n logn for any a 0 this says that m n o n log n is the largest order such that the entries of the first m n columns of n can be approximated simultaneously by independent standard normals

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

notes on the wasserstein metric in hilbert spaces let x y be a pair of hilbert valued random variables for which the wasserstein distance between the marginal distributions is reached we prove that the mapping omega rightarrow x omega y omega is increasing in a certain sense moreover if y satisfies a nondegeneration condition we can take x t y with t monotone in the sense of zarantarello we apply these results to obtain a proof of the central limit theorem clt in hilbert spaces which does not make use of the clt for real valued random variables

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

ironing sweeping and multidimensional screening we provide existence proofs and characterization results for the multidimensional version of the multiproduct monopolist problem of mussa and rosen 1978 these results are also directly applicable to the multidimensional nonlinear pricing problems studied by wilson 1993 and armstrong 1996 we establish that bunching is robust in these multidimensional screening problems even with very regular distributions of types this comes from a strong conflict between participation constraints and second order incentive compatibility conditions we consequently design a new technique the sweeping procedure for dealing with bunching in multidimensional contexts this technique extends the ironing procedure of mussa and rosen 1978 to several dimensions we illustrate it on two examples we first solve a linear quadratic version of the bidimensional nonlinear pricing problem where consumers types are exponentially distributed the solution involves pure bundling for consumers with low demands the second example is the bidimensional version of the mussa and rosen problem when consumers types are uniformly distributed on a square the solution is such that the seller offers a full differentiation of products in the upper part of the qualities spectrum but only limited choice for lower qualities this seems to be a quite general pattern for multidimensional screening problems the sweeping procedure is potentially applicable to other multidimensional screening problems

truncations of random unitary matrices we analyse properties of non hermitian matrices of size m constructed as square submatrices of unitary orthogonal random matrices of size n m distributed according to the haar measure in this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle in the limit of large matrices this ensemble is characterized by the ratio m n for the truncated cue we analytically derive the joint density of eigenvalues and all correlation functions in the strongly non unitary case universal ginibre behaviour is found for n m fixed and n to the universal resonance width distribution with n m open channels is recovered

strict convexity and c1 α regularity of potential functions in optimal transportation under condition a3w abstract in this paper we prove the strict c convexity and the c 1 regularity for potential functions in optimal transportation under condition a3w these results were obtained by caffarelli 1 3 4 for the cost c x y x y 2 by liu 11 loeper 15 trudinger and wang 20 for costs satisfying the condition a3 for costs satisfying the condition a3w the results have also been proved by figalli kim and mccann 6 assuming that the initial and target domains are uniformly c convex see also 21 and by guillen and kitagawa 8 assuming the cost function satisfies a3w in larger domains in this paper we prove the strict c convexity and the c 1 regularity assuming either the support of source density is compactly contained in a larger domain where the cost function satisfies a3w or the dimension 2 n 4

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

optimal transportation between unequal dimensions we establish that solving an optimal transportation problem in which the source and target densities are defined on spaces with different dimensions is equivalent to solving a new nonlocal analog of the monge ampere equation introduced here for the first time under suitable topological conditions we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution we show that this local equation is elliptic and c 2 alpha solutions can therefore be bootstrapped to obtain higher regularity results assuming smoothness of the corresponding differential operator which we prove under simplifying assumptions for one dimensional targets our sufficient criteria for regularity of solutions to the resulting ode are considerably less restrictive than those required by earlier works

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

improving spectral variation bounds with chebyshev polynomials a new spectral variation bound is established for linear operators a and b on finite dimensional vector spaces this bound is v a b c a b 1n a b n 1 n with c 8 this result affirmatively answers a conjecture of friedland

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

large deviation for the empirical eigenvalue density of truncated haar unitary matrices let u m be an m m haar unitary matrix and u m n be its n n truncation in this paper the large deviation is proven for the empirical eigenvalue density of u m n as m n and n the rate function and the limit distribution are given explicitly u m n is the random matrix model of quq where u is a haar unitary in a finite von neumann algebra q is a certain projection and they are free the limit distribution coincides with the brown measure of the operator quq

holder continuity and injectivity of optimal maps consider transportation of one distribution of mass onto another chosen to optimize the total expected cost where cost per unit mass transported from x to y is given by a smooth function c x y if the source density f x is bounded away from zero and infinity in an open region u subset mathbf r n and the target density f y is bounded away from zero and infinity on its support overline v subset mathbf r n which is strongly c convex with respect to u and the transportation cost c satisfies the mathbf a3 rm w condition of trudinger and wang ann sc norm super pisa cl sci 5 8 1 143 174 2009 we deduce the local holder continuity and injectivity of the optimal map inside u so that the associated potential u belongs to c 1 alpha loc u here the exponent 0 depends only on the dimension and the bounds on the densities but not on c our result provides a crucial step in the low interior regularity setting in a sequel figalli et al j eur math soc jems 1131 1166 2013 we use it to establish regularity of optimal maps with respect to the riemannian distance squared on arbitrary products of spheres three key tools are introduced in the present paper namely we first find a transformation that under mathbf a3 rm w makes c convex functions level set convex as was also obtained independently from us by liu calc var partial diff eq 34 435 451 2009 we then derive new alexandrov type estimates for the level set convex c convex functions and a topological lemma showing that optimal maps do not mix the interior with the boundary this topological lemma which does not require mathbf a3 rm w is needed by figalli and loeper calc var partial diff eq 35 537 550 2009 to conclude the continuity of optimal maps in two dimensions in higher dimensions if the densities f are holder continuous our result permits continuous differentiability of the map inside u in fact c 2 alpha loc regularity of the associated potential to be deduced from the work of liu et al comm partial diff eq 35 1 165 184 2010

elliptic pdes with constant coefficients on convex polyhedra via the unified method abstract we provide a new method to study the classical dirichlet problem for constant coefficient second order elliptic pdes on convex polyhedrons our approach is heavily motivated by fokas unified method for boundary value problems and can be interpreted as the fourier analogue to the classical boundary integral equations the central object in this approach is the global relation an integral equation which couples the known boundary data and the unknown boundary values this integral equation depends holomorphically on two complex parameters and the resulting analysis takes place on a banach space of complex analytic functions closely related to the classical paley wiener space we write the global relation in the form of an operator equation and show that the analysis can be reduced to the case of laplace s equation from which the more general problem turns out to be a compact perturbation we give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realisation of the fundamental principle of ehrenpreis for all constant coefficient elliptic pdes on convex polyhedra

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

the boundary method for semi discrete optimal transport partitions and wasserstein distance computation abstract we introduce a new technique which we call the boundary method for solving semi discrete optimal transport problems with a wide range of cost functions the boundary method reduces the effective dimension of the problem thus improving complexity for cost functions equal to a p norm with p 1 we provide mathematical justification convergence analysis and algorithmic development our testing supports the boundary method with these p norms as well as other more general cost functions

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

on the rigorous foundations of the fokas method for linear elliptic partial differential equations in this paper we address some of the rigorous foundations of the fokas method confining attention to boundary value problems for linear elliptic partial differential equations on bounded convex domains the central object in the method is the global relation which is an integral equation in the spectral fourier space that couples the given boundary data with the unknown boundary values using techniques from complex analysis of several variables we prove that a solution to the global relation provides a solution to the corresponding boundary value problem and that the solution to the global relation is unique the result holds for any number of spatial dimensions and for a variety of boundary value problems

laplace s equation on convex polyhedra via the unified method we provide a new method to study the classical dirichlet problem for laplace s equation on a convex polyhedron this new approach was motivated by fokas unified method for boundary value problems the central object in this approach is the global relation an integral equation which couples the known boundary data and the unknown boundary values this integral equation depends holomorphically on two complex parameters and the resulting analysis takes place on a banach space of complex analytic functions closely related to the classical paley wiener space we write the global relation in the form of an operator equation and prove that the relevant operator is bounded below using some novel integral identities we give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realization of the fundamental principle of ehrenpreis

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159