on the joint distribution of the maximum and its position of the airy2 process minus a parabola the maximal point of the airy2 process minus a parabola is believed to describe the scaling limit of the end point of the directed polymer in a random medium this was proved to be true for a few specific cases recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained one by moreno flores quastel and remenik and the other by schehr the first formula is given in terms of the airy function and an associated operator and the second formula is expressed in terms of the lax pair equations of the painleve ii equation we give a direct proof that these two formulas are the same

entropy and boundary conditions in random rhombus tilings the tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by fixed boundary conditions we establish a link between those conditions and free or periodic boundary ones the entropy is written as a functional integral which is treated via a saddle point method we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary this inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones this method uses a representation of tilings by membranes embedded in a higher dimensional hypercubic lattice it is illustrated in the case of 60 degree rhombus tilings

asymptotics of plancherel measures for the infinite dimensional unitary group we study a two dimensional family of probability measures on infinite gelfand tsetlin schemes induced by a distinguished family of extreme characters of the infinite dimensional unitary group these measures are unitary group analogs of the well known plancherel measures for symmetric groups we show that any measure from our family defines a determinantal point process on z x z and we prove that in appropriate scaling limits such processes converge to two different extensions of the discrete sine process as well as to the extended airy and pearcey processes

step fluctuations for a faceted crystal a statistical mechanics model for a faceted crystal is the 3d ising model at zero temperature it is assumed that in one octant all sites are occupied by atoms the remaining ones being empty allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds if v denotes the number of atoms removed then the grand canonical boltzmann weight is q v 0 q 1 as shown by cerf and kenyon in the limit q 1 a deterministic shape is attained which has the three facets 100 010 001 and a rounded piece interpolating between them we analyse the step statistics as q 1 in the rounded piece it is given by a determinantal process based on the discrete sine kernel exactly at the facet edge the steps have more space to meander their statistics is again determinantal but this time based on the airy kernel in particular the border step is well approximated by the airy process which has been obtained previously in the context of growth models our results are based on the asymptotic analysis for space time inhomogeneous transfer matrices

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

state observation problem for general time reversible system and applications this paper is concerned with a state observation problem for the general time reversible system i e to recover the initial state of the given system from suitable observation data an abstract unified framework is presented by means of an iterative time reversal technique we derive two asymptotic formulae of reconstructing the initial state approximately with explicit convergence rates general results are then applied to the state observation problems for the schrodinger equation and the wave equation

height fluctuations in the honeycomb dimer model we study a model of random surfaces arising in the dimer model on the honeycomb lattice for a fixed wire frame boundary condition as the lattice spacing 0 cohn kenyon and propp 3 showed the almost sure convergence of a random surface to a non random limit shape 0 in 12 okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions in particular constructing a natural conformal structure on them we show here that when 0 has no facets for a family of boundary conditions approximating the wire frame the large scale surface fluctuations height fluctuations about 0 converge as 0 to a gaussian free field for the above conformal structure we also show that the local statistics of the fluctuations near a given point x are as conjectured in 3 given by the unique ergodic gibbs measure on plane configurations whose slope is the slope of the tangent plane of 0 at x

shuffling algorithm for boxed plane partitions we introduce discrete time markov chains that preserve uniform measures on boxed plane partitions elementary markov steps change the size of the box from a b c to a 1 b 1 c or a 1 b 1 c algorithmic realization of each step involves o a b c operations one application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions trajectories of our markov chains can be viewed as random point configurations in the three dimensional lattice we compute the bulk limits of the correlation functions of the resulting random point process on suitable two dimensional sections the limiting correlation functions define a two dimensional determinantal point processes with certain gibbs properties

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

a periodic hexagon tiling model and non hermitian orthogonal polynomials we study a one parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling the description of the asymptotic behavior can be separated into two regimes the low and the high temperature regime our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there for low temperatures the disordered region consists of two disjoint ellipses in the high temperature regime the two ellipses merge into a single simply connected region at the transition from the low to the high temperature a tacnode appears the key to our asymptotic study is a recent approach introduced by duits and kuijlaars providing a double integral representation for the correlation kernel one of the factors in the integrand is the christoffel darboux kernel associated to polynomials that satisfy non hermitian orthogonality relations with respect to a complex valued weight on a contour in the complex plane we compute the asymptotic behavior of these orthogonal polynomials and the christoffel darboux kernel by means of a riemann hilbert analysis after substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments

operation of an associative memory abstract this paper describes the operation of an associative memory lyam governed by only ordinary differential equations useful for pattern clustering several computer simulations illustrate its operation as an unsupervised classifier vector quantizer and content addressable memory

random skew plane partitions and the pearcey process we study random skew 3d partitions weighted by qvol and specifically the q 1 asymptotics of local correlations near various points of the limit shape we obtain sine kernel asymptotics for correlations in the bulk of the disordered region airy kernel asymptotics near a general point of the frozen boundary and a pearcey kernel asymptotics near a cusp of the frozen boundary

lozenge tilings and hurwitz numbers we give a new proof of the fact that near a turning point of the frozen boundary the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a gue random matrix our argument uses none of the standard tools of integrable probability in their place it uses a combinatorial interpretation of the harish chandra itzykson zuber integral as a generating function for desymmetrized hurwitz numbers

the boundary of the gelfand tsetlin graph a new approach the gelfand tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups the boundary of the gelfand tsetlin graph has at least three incarnations as a discrete potential theory boundary as the set of finite indecomposable characters of the infinite dimensional unitary group and as the set of doubly infinite totally positive sequences an old deep result due to albert edrei and dan voiculescu provides an explicit description of the boundary it can be realized as a region in an infinite dimensional coordinate space the paper contains a novel approach to the edrei voiculescu theorem it is based on a new explicit formula for the number of semi standard young tableaux of a given skew shape or of gelfand tsetlin schemes of trapezoidal shape the formula is obtained via the theory of symmetric functions and new schur like symmetric functions play a key role in the derivation

bulk universality for random lozenge tilings near straight boundaries and for tensor products we prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains the result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in gelfand tsetlin bases of tensor products of representations of unitary groups in a weaker form our theorem also applies to random domino tilings

tilings of non convex polygons skew young tableaux and determinantal processes this paper studies random lozenge tilings of general non convex polygonal regions we show that the pairwise interaction of the non convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations the precise geometrical figure here consists of a hexagon with cuts along opposite edges for this model we take limits when the size of the hexagon and the cuts tend to infinity while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit we show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral where the number of integrations is related to the fixed geometric data above the limiting kernel is believed to be a universal master kernel

asymptotics of random lozenge tilings via gelfand tsetlin schemes a gelfand tsetlin scheme of depth n is a triangular array with m integers at level m m 1 ldots n subject to certain interlacing constraints we study the ensemble of uniformly random gelfand tsetlin schemes with arbitrary fixed n th row we obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble and also of its q deformation this provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice or equivalently of random stepped surfaces we work with a class of polygons which allows arbitrarily large number of sides we show that the local limit behavior of random tilings as all dimensions of the polygon grow is directed by ergodic translation invariant gibbs measures the slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by kenyon and okounkov acta math 199 2 263 302 2007 we also prove that at the edge of the limit shape the asymptotic behavior of random tilings is given by the airy process in particular our results cover the most investigated case of random boxed plane partitions when the polygon is a hexagon

limit shapes and local statistics for the stochastic six vertex model in this paper we consider the stochastic six vertex model on a cylinder with arbitrary initial data first we show that it exhibits a limit shape in the thermodynamic limit whose density profile is given by the entropy solution to an explicit non linear conservation law that was predicted by gwa spohn phys rev lett 68 725 728 1992 and by reshetikhin sridhar commun math phys 363 741 765 2018 then we show that the local statistics of this model around any continuity point of its limit shape are given by an infinite volume translation invariant gibbs measure of the appropriate slope

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

random sorting networks local statistics via random matrix laws this paper finds the bulk local limit of the swap process of uniformly random sorting networks the limit object is defined through a deterministic procedure a local version of the edelman greene algorithm applied to a two dimensional determinantal point process with explicit kernel the latter describes the asymptotic joint law near 0 of the eigenvalues of the corners in the antisymmetric gaussian unitary ensemble in particular the limiting law of the first time a given swap appears in a random sorting network is identified with the limiting distribution of the closest to 0 eigenvalue in the antisymmetric gue moreover the asymptotic gap in the bulk between appearances of a given swap is the gaudin mehta law the limiting universal distribution for gaps between eigenvalues of real symmetric random matrices the proofs rely on the determinantal structure and a double contour integral representation for the kernel of random poissonized young tableaux of arbitrary shape

domino statistics of the two periodic aztec diamond random domino tilings of the aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory as a statistical mechanical model it can be thought of as a dimer model or as a certain random surface we consider the aztec diamond with a two periodic weighting which exhibits all three possible phases that occur in these types of models often referred to as solid liquid and gas to analyze this model we use entries of the inverse kasteleyn matrix which give the probability of any configuration of dominoes a formula for these entries for this particular model was derived by chhita and young 2014 in this paper we find a major simplification of this formula expressing entries of the inverse kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics in a part of the aztec diamond where the asymptotic analysis is simpler we use this formula to show that the entries of the inverse kasteleyn matrix converge to the known entries of the full plane inverse kasteleyn matrices for the different phases we also study the detailed asymptotics of the inverse kasteleyn matrix at both the liquid solid and liquid gas boundaries and find the extended airy kernel in the next order asymptotics finally we provide a potential candidate for a combinatorial description of the liquid gas boundary

limit shapes and the complex burgers equation in this paper we study surfaces in r 3 that arise as limit shapes in random surface models related to planar dimers these limit shapes are surface tension minimizers that is they minimize a functional of the form h dx dy among all lipschitz functions h taking given values on the boundary of the domain the surface tension has singularities and is not strictly convex which leads to formation of facets and edges in the limit shapes we find a change of variables that reduces the euler lagrange equation for the variational problem to the complex inviscid burgers equation complex hopf equation the equation can thus be solved in terms of an arbitrary holomorphic function which is somewhat similar in spirit to weierstrass parametrization of minimal surfaces we further show that for a natural dense set of boundary conditions the holomorphic function in question is in fact algebraic the tools of algebraic geometry can thus be brought in to study the minimizers and especially the formation of their singularities this is illustrated by several explicitly computed examples

asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory we develop a new method for studying the asymptotics of symmetric polynomials of representation theoretic origin as the number of variables tends to infinity several applications of our method are presented we prove a number of theorems concerning characters of infinite dimensional unitary group and their q deformations we study the behavior of uniformly random lozenge tilings of large polygonal domains and find the gue eigenvalues distribution in the limit we also investigate similar behavior for alternating sign matrices equivalently six vertex model with domain wall boundary conditions finally we compute the asymptotic expansion of certain observables in theo n 1 dense loop model

configurational entropy of codimension one tilings and directed membranes the calculation of random tiling configurational entropy amounts to an enumeration of partitions a geometrical description of the configuration space is given in terms of integral points in a high dimensional space and the entropy is deduced from the integral volume of a convex polytope in some cases the latter volume can be expressed in a compact multiplicative formula and in all cases in terms of binomial series the origin of which is given a geometrical meaning our results mainly concern codimension one tilings but can also be extended to higher codimension tilings we also discuss the link between freeboundary and fixed boundary condition problems

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn