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Mark Adler

Publications and source records attributed to Mark Adler.

At least 19 recordsLinked to original sources
arXiv2026

Transversal Cusp-Airy versus Cusp-Airy for Lozenge Tilings

The fluctuations of lozenge tilings of hexagons with one or several cuts (nonconvexities) along opposite sides are governed by the (discrete-continuous) tacnode kernel ${\mathbb L}^{\mbox{\tiny dTac}}$, upon letting the hexagon become very large (or in other terms, keeping the hexagon fixed, with the tiles becoming very small). This is a point process with a finite number $r$ of (continuous) points along a discrete set of parallel lines within a specific region (see \cite{AJvM1,AJvM2}). Letting $r\to\infty$, one finds a liquid phase inscribed in the polygon, whose boundary (arctic curve) has a cusp near each cut, with two solid phases descending into the cusp (split-cusp). Duse-Johansson-Metcalfe \cite{DJM} show that in this situation the tile-fluctuations should obey the cusp-Airy statistics. It would have seem natural to expect to see the same cusp-Airy kernel in the neighborhood of the cut, for the limit ($r\to \infty$) of the tacnode kernel ${\mathbb L}^{\mbox{\tiny dTac}}$. As it turns out, another statistics appears: the {\em transversal cusp-Airy} statistics, which was a puzzling fact to all of us. This statistics is derived and fully explained in this paper.

math-ph↗
arXiv2023

Double Interlacing in Random Tiling Models

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.

math-ph↗
arXiv2019

A singular Toeplitz determinant and the discrete tacnode kernel for skew-Aztec Rectangles

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.

math-ph↗
arXiv2018

Probability distributions related to tilings of non-convex Polygons

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.

math-ph↗
arXiv2018

The AKS Theorem, A.C.I. Systems and Random Matrix Theory

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.

math-ph↗
arXiv2016

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 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.

math-ph↗
arXiv2013

Coupled GUE-minor Processes

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.

math.PR↗
arXiv2013

Nonintersecting random walks in the neighborhood of a symmetric tacnode

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 $m=\lfloor2t+σt^{1/3}\rfloor$, where parameter $σ$ controls the strength of interaction between the two groups of random walkers.

math-ph↗
arXiv2013

Random matrix minor processes related to percolation theory

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.

math.PR↗
arXiv2013

Tacnode GUE-minor Processes and Double Aztec Diamonds

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.

math.PR↗
arXiv2012

Consecutive Minors for Dyson's Brownian Motions

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 β=2,~4.

math.PR↗
arXiv2012

The Dyson Brownian minor process

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.

math.PR↗
arXiv2011

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 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.

math.PR↗
arXiv2011

Non-intersecting Brownian motions leaving from and going to several points

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 $0<t<1$, 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 $E=\cup_{i=1}^{r}[c_{2i-1},c_{2i}]\subset\mathbb{R}$ 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.

math.PR↗
arXiv2010

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 $\sqrt{2n}$, 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.

math.PR↗
arXiv2010

From the Pearcey to the Airy process

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.

math.PR↗
arXiv2009

A PDE for Nonintersecting Brownian Motions and Applications

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.

math.PR↗