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Gourab Ray

Publications and source records attributed to Gourab Ray.

35 records · Page 2Linked to original sources
arXiv2020

Proper 3-colorings of $\mathbb{Z}^2$ are Bernoulli

We consider the unique measure of maximal entropy for proper 3-colorings of $\mathbb{Z}^2$, or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on $\mathbb{Z}^2$. Along the way, we obtain various estimates on the mixing properties of this measure.

math.PR↗
arXiv2019

A characterisation of the Gaussian free field

We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.

math.PR↗
arXiv2019

On Testing for Parameters in Ising Models

We consider testing for the parameters of Ferromagnetic Ising models. While testing for the presence of possibly sparse magnetizations, we provide a general lower bound of minimax separation rates which yields sharp results in high temperature regimes. Our matching upper bounds are adaptive over both underlying dependence graph and temperature parameter. Moreover our results include the nearest neighbor model on lattices, the sparse Erdös-Rényi random graphs, and regular rooted trees -- right up to the critical parameter in the high temperature regime. We also provide parallel results for the entire low temperature regime in nearest neighbor model on lattices -- however in the plus boundary pure phase. Our results for the nearest neighbor model crucially depends on finite volume analogues of correlation decay property for both high and low temperature regimes -- the derivation of which borrows crucial ideas from FK-percolation theory and might be of independent interest. Finally, we also derive lower bounds for estimation and testing rates in two parameter Ising models -- which turn out to be optimal according to several recent results in this area.

math.ST↗
arXiv2019

A short proof of the discontinuity of phase transition in the planar random-cluster model with $q>4$

The goal of this paper is to provide a short proof of the discontinuity of phase transition for the random-cluster model on the square lattice with parameter $q>4$. This result was recently shown via the so-called Bethe ansatz for the six-vertex model. Our proof also exploits the connection to the six-vertex model, but does not rely on the Bethe ansatz. Our argument is soft and only uses very basic properties of the random-cluster model (for example, we do not need the Russo--Seymour--Welsh theory).

math.PR↗
arXiv2018

Dimers and Imaginary geometry

We present a general result which shows that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds true assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. This includes the case of lozenge tilings with boundary conditions lying in a plane, and Temperleyan domains in isoradial graphs (recovering a recent result of Li). The robustness of our approach, which is a key novelty of this paper, comes from the fact that the exactly solvable nature of the model plays only a minor role in the analysis. Instead, we rely on a connection to imaginary geometry, where the limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field.

math.PR↗
arXiv2018

Hyperbolic and Parabolic Unimodular Random Maps

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.

math.PR↗
arXiv2016

A note on dimers and T-graphs

The purpose of this note is to give a succinct summary of some basic properties of T-graphs which arise in the study of the dimer model. We focus in particular on the relation between the dimer model on the heaxgonal lattice with a given slope, and the behaviour of the uniform spanning tree on the associated T-graph. Together with the main result of the companion paper \cite{BLR16}, the results here show Gaussian free field fluctuations for the height function in some dimer models.

math.PR↗
arXiv2016

Unimodular Hyperbolic Triangulations: Circle Packing and Random Walk

We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.

math.PR↗
arXiv2016

Classification of scaling limits of uniform quadrangulations with a boundary

We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $θ$ and the infinite-volume Brownian disk of perimeter $σ$. We also obtain various coupling and limit results clarifying the relation between these objects.

math.PR↗
arXiv2016

Critical exponents on Fortuin--Kasteleyn weighted planar maps

In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be $$ \frac{4}π \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{κ'}{8}. $$ where $κ' $ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop which is shown to be 1 for all values of $q \in (0,4)$. Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.

math.PR↗
arXiv2016

The half plane UIPT is recurrent

We prove that the half plane version of the uniform infinite planar triangulation (UIPT) is recurrent. The key ingredients of the proof are a construction of a new full plane extension of the half plane UIPT, based on a natural decomposition of the half plane UIPT into independent layers, and an extension of previous methods for proving recurrence of weak local limits (still using circle packings).

math.PR↗
arXiv2014

Random walks on stochastic hyperbolic half planar triangulations

We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in Angel and Ray [3]. We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like $\exp(-cn^{1/3})$.

math.PR↗
arXiv2014

Geometry and percolation on half planar triangulations

We analyze the geometry of domain Markov half planar triangulations. In \cite{AR13} it is shown that there exists a one-parameter family of measures supported on half planar triangulations satisfying translation invariance and domain Markov property. We study the geometry of these maps and show that they exhibit a sharp phase-transition in view of their geometry at $α= 2/3$. For $α<2/3$, the maps form a tree-like stricture with infinitely many small cut-sets. For $α> 2/3$, we obtain maps of hyperbolic nature with exponential growth and anchored expansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.

math.PR↗
arXiv2014

Large Unicellular maps in high genus

We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges of the map. We prove that the distance between two uniformly selected vertices of such a map is of order $\log n$ and the diameter is also of order $\log n$ with high probability. We further prove that the map is locally planar with high probability. The main ingredient of the proofs is an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy.

math.PR↗
arXiv2014

Classification of Half Planar Maps

We characterize all translation invariant half planar maps satisfying a certain natural domain Markov property. For p-angulations with p \ge 3 where all faces are simple, we show that these form a one-parameter family of measures H^{(p)}_{alpha}. For triangulations we also establish existence of a phase transition which affects many properties of these maps. The critical maps are the well known half plane uniform infinite planar maps. The sub-critical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.

math.PR↗
arXiv2013

The local limit of unicellular maps in high genus

We show that the local limit of unicellular maps whose genus is proportional to the number of edges is a supercritical geometric Galton-Watson tree conditioned to survive. The proof relies on enumeration results obtained via the recent bijection given by the second author together with Feray and Fusy.

math.PR↗