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Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid-liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain-boundary, the tiles of a certain shape form for large-size domains a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a GUE-matrix (Gaussian unitary ensemble). Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: they can merely touch, leading to either a split tacnode (also called hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the nonconvex domains and probing about such split tacnodes, filaments of tiles of a certain type will connect the liquid patches: they evolve in a bricklike-sea of dimers of another type. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel; i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables one to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino-tilings of skew-Aztec rectangles and (ii) lozenge-tilings of hexagons with cuts along opposite edges. Soft, opposed to hard, tacnodes appear when two arctic curves gently touch each other amidst a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience., Comment: 67 pages

Random tilings of geometrical shapes with dominos or lozenges have been a rich source of universal statistical distributions. This paper deals with domino tilings of checker board rectangular shapes such that the top two and bottom two adjacent squares have the same orientation and the two most left and two most right ones as well. It forces these so-called "skew-Aztec rectangles" to have cuts on either side. For large sizes of the domain and upon an appropriate scaling of the location of the cuts, one finds split tacnodes between liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, filaments appear between the liquid patches evolving in a bricklike sea of dimers of another type. This work shows that the random fluctuations in a neighborhood of the split tacnode are governed asymptotically by the discrete tacnode kernel, providing strong evidence that this kernel is a universal discrete-continuous limiting kernel occurring naturally whenever we have double interlacing patterns. The analysis involves the inversion of a singular Toeplitz matrix which leads to considerable difficulties., Comment: 79 pages, 8 figures

This paper is based on the study of random lozenge tilings of non-convex polygonal regions with interacting non-convexities (cuts) and the corresponding asymptotic kernel as in [3] and [4] (discrete tacnode kernel). Here this kernel is used to find the probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts. These distributions are new., Comment: 32 pages, 4 figures

This paper gives the most general form of the Adler-Kostant-Symes Theorem, and many applications of it, both finite and infinite dimensional, the former yielding algebraic completely integrable (a.c.i.) systems, and the latter examples in random matrix theory., Comment: Topical Review (special volume 50 years Toda Lattice), with typos corrected and some additional references

This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties., Comment: 62 pages, 25 Figures

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 interaction beyond 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., Comment: 60 pages, 8 figures Some references added in section 1

This paper deals with two GUE-matrices, coupled together through some inequalities between the spectra of the first few (small) principal minors. The main results of the paper is to show that the spectra of the principal minors of these coupled matrices behave statistically as the domino tilings of finitely overlapping Aztec diamonds when their sizes get very large, with horizontal and vertical dominos being equally likely. This extends naturally a result of Johansson and Nordenstam in [17], stating that the spectra of the principal minors of a GUE-matrix behave statistically as domino tilings of an Aztec diamond, near the middle of its edge. Given the spectra of the two coupled matrices, the joint spectra of the underlying principal minors of the two GUE-matrices are uniformly distributed in a certain "double cone". In particular, this leads to two GUE-matrices sharing the same real line, with one spectrum being completely to the left of the other spectrum; this gives a new and simple extension of GUE. Also notice that all statements concerning these coupled random matrices have a domino tiling counterpart., Comment: 56 pages, 10 figures

We study random domino tilings of a Double Aztec diamond, a region consisting of two overlapping Aztec diamonds. The random tilings give rise to two discrete determinantal point processes called the K-and L-particle processes. The correlation kernel of the K-particles was derived in Adler, Johansson and van Moerbeke (2011), who used it to study the limit process of the K-particles with different weights for horizontal and vertical dominos. Let the size of both, the Double Aztec diamond and the overlap, tend to infinity such that the two arctic ellipses just touch; then they show that the fluctuations of the K-particles near the tangency point tend to the tacnode process. In this paper, we find the limiting point process of the L-particles in the overlap when the weights of the horizontal and vertical dominos are equal, or asymptotically equal, as the Double Aztec diamond grows, while keeping the overlap finite. In this case the two limiting arctic circles are tangent in the overlap and the behavior of the L-particles in the vicinity of the point of tangency can then be viewed as two colliding GUE-minor process, which we call the tacnode GUE minor process. As part of the derivation of the kernel for the L-particles we find the inverse Kasteleyn matrix for the dimer model version of Double Aztec diamond.

This paper studies a number of matrix models of size n and the associated Markov chains for the eigenvalues of the models for consecutive n's. They are consecutive principal minors for two of the models, GUE with external source and the multiple Laguerre matrix model, and merely properly defined consecutive matrices for the third one, the Jacobi-Pineiro model; nevertheless the eigenvalues of the consecutive models all interlace. We show: (i) For each of those finite models, we give the transition probability of the associated Markov chain and the joint distribution of the entire interlacing set of eigenvalues; we show this is a determinantal point process whose extended kernels share many common features. (ii) To each of these models and their set of eigenvalues, we associate a last-passage percolation model, either finite percolation or percolation along an infinite strip of finite width, yielding a precise relationship between the last passage times and the eigenvalues. (iii) Finally it is shown that for appropriate choices of exponential distribution on the percolation, with very small means, the rescaled last passage times lead to the Pearcey process; this should connect the Pearcey statistics with random directed polymers., Comment: 57 pages, 15 figures; more discussion on the relation to percolation and directed polymer; some more references added

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 been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the "tacnode process". It is also shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily., Comment: 80 pages, 18 figures

Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on the real line for the eigenvalues, as was discovered by Dyson. Applying scaling limits to the random matrix models, combined with Dyson's dynamics, then leads to interesting, infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between two of the models, namely the Airy and Pearcey processes and more precisely shows how to approximate the multi-time statistics for the Pearcey process by the one of the Airy process with the help of a PDE governing the gap probabilities for the Pearcey process., Comment: 21 pages, 2 figures

Consider a continuous time random walk in $\mathbb{Z}$ with independent and exponentially distributed jumps $\pm1$. The model in this paper consists in an infinite number of such random walks starting from the complement of $\{-m,-m+1,\ldots,m-1,m\}$ 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 $m\simeq2t$ to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling $m=\lfloor2t+\sigma t^{1/3}\rfloor$, where parameter $\sigma$ controls the strength of interaction between the two groups of random walkers., Comment: Published in at http://dx.doi.org/10.1214/11-AOP726 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In 1962, Dyson introduced dynamics in random matrix models, in particular into GUE (also for beta=1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: beta=1,~2,~4. It is also shown that the spectra of three consecutive minors ceases to be Markovian for \beta=2,~4., Comment: 32 pages, 1 figure

Consider an $n\times n$ Hermitean matrix valued stochastic process $\{H_t\}_{t\geq 0}$ where the matrix elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k\times k$ in the upper left corner of $H_t$. If you project this process to a space-like path it is a determinantal process and we compute the kernel. This kernel contains as special cases the well known GUE minor kernel, discovered by Johansson-Nordenstam and Okounkov-Reshetikhin in 2006, and the Dyson Brownian motion kernel discovered by Forrester-Nagao in 1998. In the bulk scaling limit of this kernel it is possible to recover a time-dependent generalisation of Boutillier's bead kernel., Comment: 30 pages, 2 figures

Consider n non-intersecting Brownian motions on $\mathbb{R}$, depending on time $t \in [0,1]$, with $m_i$ particles forced to leave from $a_i$ at time $t=0$, $1\leq i\leq q$, and $n_j$ particles forced to end up at $b_j$ at time $t=1$, $1\leq j\leq p$. 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

Consider non-intersecting Brownian motions on the real line, starting from the origin at t=0, with a number of particles forced to reach p distinct target points at time t=1. This work shows that the transition probability, that is the probability for the particles to pass through windows E_k at times t_k, satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p+1, with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations from a more probabilistic point of view. As an application of these equations, let the number of particles forced to the extreme target points (the first and the last one) tend to infinity; keep the number of particles forced to intermediate target points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to the first target point depart from those going to the last target point. These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. Finally, the paper contains a conjecture., Comment: 44 pages, 2 figures

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.

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 $\sqrt{2n}$, 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., Comment: Published in at http://dx.doi.org/10.1214/09-AOP493 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Consider n non-intersecting particles on the real line (Dyson Brownian motions), all starting from the origin at time=0, and forced to return to x=0 at time=1. For large n, the average mean density of particles has its support, for each 0

Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of time deformations for each weight, the multi-component KP-hierarchy: these determinants are thus "tau-functions" for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. As an application, the multi-component KP-hierarchy leads to a large set of non-linear PDE's, which are useful in finding partial differential equations for the transition probabilities of certain infinite-dimensional diffusions., Comment: 37 pages, 1 figure

This paper is a step in the direction of understanding the behavior of non-intersecting Brownian motions on the real line, when the number of particles becomes large. Consider 2k non-intersecting Brownian motions, all starting at the origin, such that the k left paths end up at -a and the k right paths end up at +a at time t=1. The Karlin-McGregor formula enables one to express the transition probability in terms of a matrix model, consisting of Gaussian Hermitian random matrices in a chain with external source. It is shown that the log of the probability for this model satisfies a fourth order PDE with a quartic non-linearity, obtained by means of the 3-component KP hierarchy and Virasoro constraints. When the number of particles grows very large, the particles will be concentrated on two intervals near t=0 and on one interval near t=1. The Pearcey process is the infinite-dimensional diffusion, near the critical transition from two to one interval. An appropriate scaling limit of the PDE for the finite model leads to a non-linear PDE for the multi-time transition probabilities of the Pearcey process. We conjecture that each of the Markov clouds (like the Pearcey process) arising near phase transitions is related to some integrable system. Moreover, there is an intimate connection between the integrable system and the associated Riemann-Hilbert problem., Comment: 56 pages

The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues a and -a. As a first result, the probability that the eigenvalues of the ensemble belong to a set satisfies a fourth order PDE with quartic non-linearity; the variables being the eigenvalue a and the boundary points of the set. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap; i.e., when the support of the equilibrium measure for large size random matrices has a gap, which can be made to close. Precisely, the Gaussian Hermitian random matrix ensemble with external source has this feature. In this work, we show the Pearcey distribution satisfies a a fourth order PDE with cubic non-linearity. The PDE for the finite problem is found by by showing that an appropriate integrable deformation of the random matrix ensemble with external source satisfies the three-component KP equation and Virasoro constraints., Comment: 33 pages

Given uniform probability on words of length M=Np+k, from an alphabet of size p, consider the probability that a word (i) contains a subsequence of letters (p, p-1,...,1) in that order and (ii) that the maximal length of the disjoint union of p-1 increasing subsequences of the word is \leq M-N . A generating function for this probability has the form of an integral over the Grassmannian of p-planes in complex C^n. The present paper shows that the asymptotics of this probability, when N tends to infinity, is related to the kth moment of the chi^2-distribution of parameter 2p^2. This is related to the behavior of the integral over the Grassmannian Gr(p,C^n) of p-planes in C^n, when the dimension of the ambient space C^n becomes very large. A different scaling limit for the Poissonized probability is related to a new matrix integral, itself a solution of the Painlev\'e IV equation. This is part of a more general set-up related to the Painlev\'e V equation., Comment: 31 pages, 1 figure

In a celebrated paper, Dyson shows that the spectrum of an n\times n random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy process at two different times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process, which itself is based on the joint probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE for the Dyson process is then subjected to an asymptotic analysis, consistent with the edge and bulk rescalings. The PDEs enable one to compute the asymptotic behavior of the joint distribution and the correlation for these processes at different times t_1 and t_2, when t_2-t_1\to \infty, as illustrated in this paper for the Airy process. This paper also contains a rigorous proof that the extended Hermite kernel, governing the joint probabilities for the Dyson process, converges to the extended Airy and Sine kernels after the appropriate rescalings., Comment: Published at http://dx.doi.org/10.1214/009117905000000107 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy Process. The latter is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles get large, with space and time appropriately rescaled. The question reduces to an asymptotic analysis on the equation governing the joint probability of the eigenvalues of coupled Gaussian Hermitian matrices. The differential equations lead to the asymptotic behavior of the joint distribution and the correlation for the Airy process at different times t_1 and t_2, when t_2-t_1 tends to infinity., Comment: 9 pages

The finite Pfaff lattice is given by commuting Lax pairs involving a finite matrix L (zero above the first subdiagonal) and a projection onto Sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t_1,t_2,..., after conjugation by a diagonal matrix. The sequence of polynomial tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of tau-functions is given inductively by the action of a fixed vertex operator. As examples, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial., Comment: 57 pages