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Kurt Johansson

Publications and source records attributed to Kurt Johansson.

At least 37 records · Page 2Linked to original sources
arXiv2017

Airy point process at the liquid-gas boundary

Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

math.PR↗
arXiv2017

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↗
arXiv2017

Edge fluctuations of limit shapes

In random tiling and dimer models we can get various limit shapes which gives the boundaries between different types of phases. The shape fluctuations at these boundaries give rise to universal limit laws, in particular the Airy process. We survey some models which can be analyzed in detail based on the fact that they are determinantal point processes with correlation kernels that can be computed. We also discuss which type of limit laws that can be obtained.

math.PR↗
arXiv2016

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.

math.PR↗
arXiv2016

Two time distribution in Brownian directed percolation

In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants which is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.

math-ph↗
arXiv2015

The Cusp-Airy Process

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.

math.PR↗
arXiv2015

Asymptotic domino statistics in the Aztec diamond

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

math.PR↗
arXiv2013

On mesoscopic equilibrium for linear statistics in Dyson's Brownian Motion

In this paper we study mesoscopic fluctuations for Dyson's Brownian motion with $β=2$. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. In this paper we investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that we consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but we have not yet reached equilibrium at the macrosopic scale. We describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. We consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, we obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.

math-ph↗
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

Non-colliding Brownian Motions and the extended tacnode process

We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.

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

Universality for certain Hermitian Wigner Matrices under weak moment conditions

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

math.PR↗
arXiv2010

Eigenvalues of GUE Minors

Consider an infinite random matrix $H=(h_{ij})_{0<i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $μ^i_j$. We show that the configuration of all these eigenvalues $(i,μ_j^i)$ form a determinantal point process on $\mathbb{N}\times\mathbb{R}$. Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.

math.PR↗
arXiv2007

A multi-dimensional Markov chain and the Meixner ensemble

We show that the transition probability of the Markoc chain $(G(j,1),...,G(j,n))_{j\ge 1}$, where the $G(i,j)'s$ are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for $G(m,n)$. We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.

math.PR↗
arXiv2007

Powers of large random unitary matrices and Toeplitz determinants

We study the limiting behavior of $\Tr U^{k(n)}$, where $U$ is a $n\times n$ random unitary matrix and $k(n)$ is a natural number that may vary with $n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence to this result, we find that for each fixed $m\in \N$, the random variables $ \Tr U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,..., m$, converge to independent standard complex normals.

math-ph↗
arXiv2007

On some special directed last-passage percolation models

We investigate extended processes given by last-passage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to non-universal and trivial extended processes.

math.PR↗