Showing papers by "Bálint Virág published in 2021"
TL;DR: In this article, it was shown that the law of the KPZ fixed point starting from arbitrary initial condition is continuous with respect to the Brownian motion B on every compact interval.
Abstract: We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion B on every compact interval. In particular, the Airy1 process is absolutely continuous with respect to B on any compact interval.
13 citations
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TL;DR: In this article, the authors provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class, based on the notion of directed metric, a generalization of metrics which behaves better under limits.
Abstract: We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence to the Airy line ensemble implies convergence to the Airy sheet. In i.i.d. environments, we show that Airy sheet convergence implies convergence of distances and geodesics to their counterparts in the directed landscape. Our results imply convergence of classical last passage models and interacting particle systems. Our framework is built on the notion of a directed metric, a generalization of metrics which behaves better under limits. As a consequence of our results, we present a solution to an old problem: the scaled longest increasing subsequence in a uniform permutation converges to the directed geodesic.
11 citations
TL;DR: It is shown that this process is the microscopic scaling limit in the bulk of the Hermite, and bounds on the variance of the point counting of the circular and the Gaussian beta ensembles are proven.
Abstract: The bead process introduced by Boutillier is a countable interlacing of the $${\text {Sine}}_2$$
point processes. We construct the bead process for general $${\text {Sine}}_{\beta }$$
processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite $$\beta $$
corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian Unitary and Orthogonal Ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circular and the Gaussian beta ensembles, proven in a companion paper (Najnudel and Virag in Some estimates on the point counting of the Circular and the Gaussian Beta Ensemble, 2019).
10 citations
TL;DR: In this paper, the authors provide a set of tools, which allow for precise probabilistic analysis of the Airy line ensemble, including independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines.
Abstract: The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools, which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson’s Brownian motion converging to the Airy line ensemble.
4 citations
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TL;DR: In this article, a version of the RSK correspondence based on the Pitman transform and geometric considerations is presented, which is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models.
Abstract: We present a version of the RSK correspondence based on the Pitman transform and geometric considerations. This version unifies ordinary RSK, dual RSK and continuous RSK. We show that this version is both a bijection and an isometry, two crucial properties for taking limits of last passage percolation models.
We use the bijective property to give a non-computational proof that dual RSK maps Bernoulli walks to nonintersecting Bernoulli walks.
2 citations
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TL;DR: In this paper, it was shown that large deviations of the interchange process are controlled by the Dirichlet energy, which implies the Archimedean limit for relaxed sorting networks and allow us to asymptotically count such networks.
Abstract: We use the framework of permuton processes to show that large deviations of the interchange process are controlled by the Dirichlet energy. This establishes a rigorous connection between processes of permutations and one-dimensional incompressible Euler equations. While our large deviation upper bound is valid in general, the lower bound applies to processes corresponding to incompressible flows, studied in this context by Brenier. These results imply the Archimedean limit for relaxed sorting networks and allow us to asymptotically count such networks.
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TL;DR: In this paper, lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear and quadratic mother groups on the orbit of the zero ray were given.
Abstract: We give lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear (degree 1) and quadratic (degree 2) mother groups on the orbit of the zero ray. These bounds, combined with results of \cite{JNS} show that every quadratic activity automaton group is amenable. The resistance bounds use an apparently new "weighted" version of the Nash-Williams criterion which may be of independent interest.