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

stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

stochastic integrators with stationary independent increments lipschitzian stochastic differential equations driven by a process with stationary independent increments permit a priori growth and stability estimates up to any sure time and the solutions depend differentiably on parameters provided the levy measure of the driving term has suitable moments

nonintersecting random walks in the neighborhood of a symmetric tacnode consider a continuous time random walk in z with independent and exponentially distributed jumps 1 the model in this paper consists in an infinite number of such random walks starting from the complement of m m 1 m 1 m at time t returning to the same starting positions at time t and conditioned not to intersect this yields a determinantal process whose gap probabilities are given by the fredholm determinant of a kernel thus this model consists of two groups of random walks which are contained within two ellipses which with the choice m 2t to leading order just touch so we have a tacnode we determine the new limit extended kernel under the scaling m 2t t1 3 where parameter controls the strength of interaction between the two groups of random walkers

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

limiting distribution of maximal crossing and nesting of poissonized random matchings the notion of r crossing and r nesting of a complete matching was introduced and a symmetry property was proved by chen et al trans amer math soc 359 2007 1555 1575 we consider random matchings of large size and study their maximal crossing and their maximal nesting it is known that the marginal distribution of each of them converges to the goe tracy widom distribution we show that the maximal crossing and the maximal nesting becomes independent asymptotically and we evaluate the joint distribution for the poissonized random matchings explicitly to the first correction term this leads to an evaluation of the asymptotic of the covariance furthermore we compute the explicit second correction term in the distribution function of two objects a the length of the longest increasing subsequence of poissonized random permutation and b the maximal crossing and hence also the maximal nesting of poissonized random matching

hierarchical spectra in systems with incommensurate potentials the energy spectrum of a potential consisting of an array of barriers is obtained by solving the schrodinger equation the barriers have the same height and constant separation but their widths are incommensurately modulated the hierarchical band splittings obey the same rules which were derived earlier for the tight binding aubry model the authors have also demonstrated their equivalence to hofstadter s rules which were derived by stinchcombe and bell 1987

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

airy processes with wanderers and new universality classes consider n m nonintersecting brownian bridges with n of them leaving from 0 at time t 1 and returning to 0 at time t 1 while the m remaining ones wanderers go from m points a i to m points b i first we keep m fixed and we scale a i b i appropriately with n in the large n limit we obtain a new airy process with wanderers in the neighborhood of root 2n the approximate location of the rightmost particle in the absence of wanderers this new process is governed by an airy type kernel with a rational perturbation letting the number m of wanderers tend to infinity as well leads to two pearcey processes about two cusps a closing and an opening cusp the location of the tips being related by an elliptic curve upon tuning the starting and target points one can let the two tips of the cusps grow very close this leads to a new process which might be governed by a kernel represented as a double integral involving the exponential of a quintic polynomial in the integration variables

dyson s nonintersecting brownian motions with a few outliers consider n nonintersecting brownian particles on r dyson brownian motions all starting from the origin at time t 0 and forced to return to x 0 at time t 1 for large n the average mean density of particles has its support for each 0 0 does it affect the brownian fluctuations along the curve c for large n in this paper we show that no new process appears as long as one considers points y t is an element of c such that 0 infinity this kernel is closely related to one found by baik ben arous and peche in the context of multivariate statistics c 2008 wiley periodicals inc

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

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

schur dynamics of the schur processes abstract we construct discrete time markov chains that preserve the class of schur processes on partitions and signatures one application is a simple exact sampling algorithm for q volume distributed skew plane partitions with an arbitrary back wall another application is a construction of markov chains on infinite gelfand tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite dimensional unitary group

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

generalized ablowitz ladik hierarchy in topological string theory this paper addresses the issue of integrable structures in topological string theory on generalized conifolds open string amplitudes of this theory can be expressed as the matrix elements of an operator on the fock space of 2d charged free fermion fields the generating function of these amplitudes with respect to the product of two independent schur functions becomes a tau function of the 2d toda hierarchy the associated lax operators turn out to have a particular factorized form this factorized form of the lax operators characterizes a generalization of the ablowitz ladik hierarchy embedded in the 2d toda hierarchy the generalized ablowitz ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

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

thermodynamic limit of random partitions and dispersionless toda hierarchy we study the thermodynamic limit of random partition models for the instanton sum of 4d and 5d supersymmetric u 1 gauge theories deformed by some physical observables the physical observables correspond to external potentials in the statistical model the partition function is reformulated in terms of the density function of maya diagrams the thermodynamic limit is governed by a limit shape of young diagrams associated with dominant terms in the partition function the limit shape is characterized by a variational problem which is further converted to a scalar valued riemann hilbert problem this riemann hilbert problem is solved with the aid of a complex curve which may be thought of as the seiberg witten curve of the deformed u 1 gauge theory this solution of the riemann hilbert problem is identified with a special solution of the dispersionless toda hierarchy that satisfies a pair of generalized string equations the generalized string equations for the 5d gauge theory are shown to be related to hidden symmetries of the statistical model the prepotential and the seiberg witten differential are also considered

a generalization of macmahon s formula we generalize the generating formula for plane partitions known as macmahon s formula as well as its analog for strict plane partitions we give a 2 parameter generalization of these formulas related to macdonald s symmetric functions the formula is especially simple in the hall littlewood case we also give a bijective proof of the analog of macmahon s formula for strict plane partitions

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

the pearcey process the extended airy kernel describes the space time correlation functions for the airy process which is the limiting process for a polynuclear growth model the airy functions themselves are given by integrals in which the exponents have a cubic singularity arising from the coalescence of two saddle points in an asymptotic analysis pearcey functions are given by integrals in which the exponents have a quartic singularity arising from the coalescence of three saddle points a corresponding pearcey kernel appears in a random matrix model and a brownian motion model for a fixed time this paper derives an extended pearcey kernel by scaling the brownian motion model at several times and a system of partial differential equations whose solution determines associated distribution functions we expect there to be a limiting nonstationary process consisting of infinitely many paths which we call the pearcey process whose space time correlation functions are expressible in terms of this extended kernel

characterizing yang mills fields by stochastic parallel transport abstract we give a new stochastic approach to yang mills fields on vector bundles by studying the derivative of the stochastic parallel transport in the vector bundle under a variation of connection we establish martingale criterions for variations transversal to the gauge orbit and for yang mills fields using variations induced by the flow of vector fields of gradient type on the base manifold in dimension four we link the local yang mills action to the quadratic variation of the derivative process defined by scalings of the driving brownian motion in a coordinate chart

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

tacnode gue minor processes and double aztec diamonds we study determinantal point processes arising in random domino tilings of a double aztec diamond a region consisting of two overlapping aztec diamonds at a turning point in a single aztec diamond where the disordered region touches the boundary the natural limiting process is the gue minor process increasing the size of a double aztec diamond while keeping the overlap between the two aztec diamonds finite we obtain a new determinantal point process which we call the tacnode gue minor process this process can be thought of as two colliding gue minor processes as part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode gue minor process we find the inverse kasteleyn matrix for the dimer model version of the double aztec diamond

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

double aztec diamonds and the tacnode process discrete and continuous non intersecting random processes have given rise to critical infinite dimensional diffusions like the airy process the pearcey process and variations thereof it has be

boxed skew plane partition and integrable phase model we study the relation between the boxed skew plane partition and the integrable phase model we introduce a generalization of a scalar product of the phase model and calculate it in two ways the first one in terms of the skew schur functions and the other one by use of the commutation relations of operators in both cases a generalized scalar product is expressed as a determinant we show that a special choice of the spectral parameters of a generalized scalar product gives the generating function of the boxed skew plane partition

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

non intersecting brownian motions leaving from and going to several points abstract consider n non intersecting brownian motions on r depending on time t 0 1 with m i particles forced to leave from a i at time t 0 1 i q and n j particles forced to end up at b j at time t 1 1 j p for arbitrary p and q it is not known if the distribution of the positions of the non intersecting brownian particles at a given time 0 t 1 is the same as the joint distribution of the eigenvalues of a matrix ensemble this paper proves the existence for general p and q of a partial differential equation pde satisfied by the log of the probability to find all the particles in a disjoint union of intervals e i 1 r c 2 i 1 c 2 i r at a given time 0 t 1 the variables are the coordinates of the starting and ending points of the particles and the boundary points of the set e the proof of the existence of such a pde using virasoro constraints and the multicomponent kp hierarchy is based on the method of elimination of the unwanted partials that this is possible is a miracle unfortunately we were unable to find its explicit expression the case p q 2 will be discussed in the last section

k theoretic boson fermion correspondence and melting crystals we study non hermitian integrable fermion and boson systems from the perspectives of grothendieck polynomials the models considered in this article are the five vertex model as a fermion system and the non hermitian phase model as a boson system both models are characterized by different solutions satisfying the same yang baxter relation from our previous works on the identification between the wavefunctions of the five vertex model and grothendieck polynomials we introduce skew grothendieck polynomials and derive the addition theorem among them using these relations we derive the wavefunctions of the non hermitian phase model as a determinant form which can also be expressed as grothendieck polynomials namely we establish a k theoretic boson fermion correspondence at the level of wavefunctions as a by product the partition function of the statistical mechanical model of a three dimensional 3d melting crystal is exactly calculated by use of the scalar products of the wavefunctions of the phase model the resultant expression can be regarded as a k theoretic generalization of the macmahon function describing the generating function of the plane partitions which interpolates the generating functions of two dimensional 2d and 3d young diagrams

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

infinite systems of noncolliding generalized meanders and riemann liouville differintegrals yor s generalized meander is a temporally inhomogeneous modification of the 2 1 dimensional bessel process with 1 in which the inhomogeneity is indexed by kappa in 0 2 nu 1 we introduce the noncolliding particle systems of the generalized meanders and prove that they are pfaffian processes in the sense that any multitime correlation function is given by a pfaffian in the infinite particle limit we show that the elements of matrix kernels of the obtained infinite pfaffian processes are generally expressed by the riemann liouville differintegrals of functions comprising the bessel functions j used in the fractional calculus where orders of differintegration are determined by as special cases of the two parameters the present infinite systems include the quaternion determinantal processes studied by forrester nagao and honner and by nagao which exhibit the temporal transitions between the universality classes of random matrix theory

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

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

random surface growth with a wall and plancherel measures for o we consider a markov evolution of lozenge tilings of a quarter plane and study its asymptotics at large times one of the boundary rays serves as a reflecting wall we observe frozen and liquid regions prove convergence of the local correlations to translation invariant gibbs measures in the liquid region and obtain new discrete jacobi and symmetric pearcey determinantal point processes near the wall the model can be viewed as the one parameter family of plancherel measures for the infinite dimensional orthogonal group and we use this interpretation to derive the determinantal formula for the correlation functions at any finite time moment

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

quasi invariance of the wiener measure on the path space over a complete riemannian manifold we prove a generalization of the cameron martin theorem for a geometrically and stochastically complete riemannian manifold namely the wiener measure on the path space over such a manifold is quasiinvariant under the flow generated by a cameron martin vector field

fractal spectra for incommensurate systems a scaling rule has been proposed to study the energy spectrum of the aubry model 1978 h sigma n v cos qn andagger an t an 1dagger an an an 1dagger an for a class of irrational numbers q 1 mu q where mu is a positive integer the scaling rule is universal in the sense that it is independent of the ratio v t and has been confirmed via a large scale numerical calculation a fractal dimensionality df of the energy spectrum is defined and has been found to be df 1

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

exact solutions for kpz type growth processes random matrices and equilibrium shapes of crystals abstract three models from statistical physics can be analyzed by employing space time determinantal processes 1 crystal facets in particular the statistical properties of the facet edge and equivalently tilings of the plane 2 one dimensional growth processes in the kardar parisi zhang universality class and directed last passage percolation 3 random matrices multi matrix models and dyson s brownian motion we explain the method and survey results of physical interest

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

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

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

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

metal insulator transition and self similarity in the aubry model the aubry model near the critical region is analysed with the numerical renormalisation group method the transition is shown to be energy dependent and the self similarity is demonstrated

airy kernel with two sets of parameters in directed percolation and random matrix theory we introduce a generalization of the extended airy kernel with two sets of real parameters we show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices

dynamics of an unbounded interface between ordered phases we investigate the evolution of a single unbounded interface between ordered phases in two dimensional ising ferromagnets that are endowed with single spin flip zero temperature glauber dynamics we examine specifically the cases where the interface initially has either one or two corners in both examples the interface evolves to a limiting self similar form we apply the continuum time dependent ginzburg landau equation and a microscopic approach to calculate the interface shape for the single corner system we also discuss a correspondence between the interface and the young diagram that represents the partition of the integers

from aztec diamonds to pyramids steep tilings we introduce a family of domino tilings that includes tilings of the aztec diamond and pyramid partitions as special cases these tilings live in a strip of mathbb z 2 of the form 1 leq x y leq 2 ell for some integer ell geq 1 and are parametrized by a binary word w in 2 ell that encodes some periodicity conditions at infinity aztec diamond and pyramid partitions correspond respectively to w ell and to the limit case w infty infty for each word w and for different types of boundary conditions we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips that admits a natural multivariate generalization the main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism which we slightly extend in order to handle littlewood type identities in probabilistic terms our tilings map to schur processes of different types standard pfaffian and periodic we also introduce a more general model that interpolates between domino tilings and plane partitions

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

critical behavior of nonintersecting brownian motions at a tacnode we study a model of n one dimensional nonintersecting brownian motions with two prescribed starting points at time t 0 and two prescribed ending points at time t 1 in a critical regime where the paths fill two tangent ellipses in the time space plane as n the limiting mean density for the positions of the brownian paths at the time of tangency consists of two touching semicircles possibly of different sizes we show that in an appropriate double scaling limit there is a new family of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new riemann hilbert problem of size 4 4 we prove solvability of the riemann hilbert problem and establish a remarkable connection with the hastings mcleod solution of the painleve ii equation we show that this painleve ii transcendent also appears in the critical limits of the recurrence coefficients of the multiple hermite polynomials that are associated with the nonintersecting brownian motions universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots which represents a new universality class 2011 wiley periodicals inc

matrix kernels for measures on partitions we consider the problem of computation of the correlation functions for the z measures with the deformation jack parameters 2 or 1 2 such measures on partitions are originated from the representation theory of the infinite symmetric group and in many ways are similar to the ensembles of random matrix theory of 4 or 1 symmetry types for a certain class of such measures we show that correlation functions can be represented as pfaffians including 2 2 matrix valued kernels and compute these kernels explicitly we also give contour integral representations for correlation kernels of closely connected measures on partitions

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

universality for the pearcey process consider non intersecting brownian motions on the line leaving from the origin and forced to two arbitrary points letting the number of brownian particles tend to infinity and upon rescaling there is a point of bifurcation where the support of the density of particles goes from one interval to two intervals in this paper we show that at that very point of bifurcation a cusp appears near which the brownian paths fluctuate like the pearcey process this is a universality result within this class of problems tracy and widom obtained such a result in the symmetric case when the two target points are symmetric with regard to the origin this asymmetry enabled us to improve considerably a result concerning the non linear partial differential equations governing the transition probabilities for the pearcey process obtained by adler and van moerbeke c 2010 elsevier b v all rights reserved

fluctuations of an atomic ledge bordering a crystalline facet when a high symmetry facet joins the rounded part of a crystal the step line density vanishes as square root of r with r denoting the distance from the facet edge this means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation we investigate the statistical properties of the border ledge fluctuations in the scaling regime they turn out to be non gaussian and related to the edge statistics of gaussian unitary ensemble multimatrix models

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

string orthogonal polynomials string equations and 2 toda symmetries 1 the 2 toda lattice and its generic symmetries 2 a larger class of symmetries for special initial conditions 3 borel decomposition of moment matrices tau functions and string orthogonal polynomials 4 from string orthogonal polynomials to the 2 toda lattice and the string equation 5 virasoro constraints on two matrix integrals comment 45 pages latex

limit processes for tasep with shocks and rarefaction fans we consider the totally asymmetric simple exclusion process tasep with two sided bernoulli initial condition i e with left density and right density we study the associated height function whose discrete gradient is given by the particle occurrences macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of we characterize the large time scaling limit of the multipoint fluctuations as a function of the densities and of the different macroscopic regions moreover using a slow decorrelation phenomena the results are extended from fixed time to the whole space time except along the some directions the characteristic solutions of the related burgers equation where the problem is still open

yang mills fields and random holonomy along brownian bridges we characterize yang mills connections in vector bundles in terms of covariant derivatives of stochastic parallel transport along variations of brownian bridges on the base manifold in particular we prove that a connection in a vector bundle e is yang mills if and only if the covariant derivative of parallel transport along brownian bridges in the direction of their drift is a local martingale when transported back to the starting point we present a taylor expansion up to order 3 for stochastic parallel transport in e along small rescaled brownian bridges and prove that the connection in e is yang mills if and only if all drift terms in the expansion up to order 3 vanish or equivalently if and only if the average rotation of parallel transport along small bridges and loops is of order 4

nearest neighbor markov dynamics on macdonald processes abstract macdonald processes are certain probability measures on two dimensional arrays of interlacing particles introduced by borodin and corwin in 7 they are defined in terms of nonnegative specializations of the macdonald symmetric functions and depend on two parameters q t 0 1 our main result is a classification of continuous time nearest neighbor markov dynamics on the space of interlacing arrays that act nicely on macdonald processes the classification unites known examples of such dynamics and also yields many new ones when t 0 one dynamics leads to a new integrable interacting particle system on the one dimensional lattice which is a q deformation of the pushtasep long range tasep when q t the macdonald processes become the schur processes of okounkov and reshetikhin 41 in this degeneration we discover new robinson schensted type correspondences between words and pairs of young tableaux that govern some of our dynamics

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

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

the quasi invariance property for the gamma kernel determinantal measure abstract the gamma kernel is a projection kernel of the form a x b y b x a y x y where a and b are certain functions on the one dimensional lattice expressed through euler s function the gamma kernel depends on two continuous parameters its principal minors serve as the correlation functions of a determinantal probability measure p defined on the space of infinite point configurations on the lattice as was shown earlier a borodin g olshanski adv math 194 2005 141 202 arxiv math ph 0305043 p describes the asymptotics of certain ensembles of random partitions in a limit regime theorem the determinantal measure p is quasi invariant with respect to finitary permutations of the nodes of the lattice this result is motivated by an application to a model of infinite particle stochastic dynamics

nonintersecting brownian bridges between reflecting or absorbing walls abstract we study a model of nonintersecting brownian bridges on an interval with either absorbing or reflecting walls at the boundaries focusing on the point in space time at which the particles meet the wall these processes are determinantal and in different scaling limits when the particles approach the reflecting resp absorbing walls we obtain hard edge limiting kernels which are the even resp odd parts of the pearcey and tacnode kernels we also show that in the single time case our hard edge tacnode kernels are equivalent to the ones studied by delvaux 16 defined in terms of a 4 4 lax pair for the inhomogeneous painleve ii equation pii as a technical ingredient in the proof we construct a schlesinger transform for the 4 4 lax pair in 16 which preserves the hastings mcleod solutions to pii

fluctuations of the maximal particle energy of the quantum ideal gas and random partitions we investigate the limiting distribution of the fluctuations of the maximal summand in a random partition of a large integer with respect to a multiplicative statistics we show that for a big family of gibbs measures on partitions so called generalized bose einstein statistics this distribution is the well known gumbel distribution which usually appears in the context of indepedent random variables in particular it means that the properly rescaled maximal energy of an individual particle in the grand canonical ensemble of the d dimensional quantum ideal gas has the gumbel distribution in the limit

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight

stochastic higher spin six vertex model and macdonald measures we prove an identity that relates the q laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a macdonald measure on the other the identity is used to prove the gue tracy widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding schur measures

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

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices