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Bálint Virág

Publications and source records attributed to Bálint Virág.

At least 19 recordsLinked to original sources
arXiv2026

The directed landscape from Brownian motion

We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This map is the natural scaling limit of the Robinson--Schensted--Knuth (RSK) correspondence. The Brownian motions arise as the marginals of the multi-path stationary horizon associated with the directed landscape. The inverse map is fully explicit and yields a natural coupling in which Brownian last-passage percolation converges in probability to the directed landscape. As an application, we prove that the directed landscape restricted to a strip can be reconstructed from the parabolic Airy line ensemble, resolving a conjecture of the first author and Zhang. Along the way we develop two new versions of RSK in the semi-discrete setting, introduce a general theory of sorting via Pitman operators that generates a faithful action of the biHecke monoid, and establish key identities for the multi-path stationary horizon for both the directed landscape and Brownian last-passage percolation.

math.PR↗
arXiv2026

The Efron-Stein inequality for identically distributed pairs

We prove that the classical Efron--Stein inequality holds for independent exchangeable pairs \((X_i,Y_i)\). The same inequality fails for independent identically distributed pairs; a simple trigonometric counterexample shows that the trivial Cauchy--Schwarz bound of factor \(n\) is sharp. When each random variable takes at most \(k_i\) values, a useful bound still holds with explicit constant \(ρ(k)\le\max_i k_i/2\).

math.PR↗
arXiv2025

The geometry of coalescing random walks, the Brownian web distance and KPZ universality

Coalescing simple random walks in the plane form an infinite tree. A natural directed distance on this tree is given by the number of jumps between branches when one is only allowed to move in one direction. The Brownian web distance is the scale-invariant limit of this directed metric. It is integer-valued and has scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world. However, we show that the shear limit of the Brownian web distance is still given by the Airy process. We conjecture that our limit theorem can be extended to the full directed landscape.

math.PR↗
arXiv2024

Three-halves variation of geodesics in the directed landscape

We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment around an interior point has a small-scale limit. This limit is given in terms of the directed landscape with Brownian-Bessel boundary conditions. The environments around different interior points are asymptotically independent. We give tail bounds with optimal exponents for geodesic and weight function increments. As an application of our results, we show that geodesics are not Hölder-$2/3$ and that weight functions are not Hölder-$1/3$, although these objects are known to be Hölder with all lower exponents.

math.PR↗
arXiv2024

Upper tail large deviations of the directed landscape

Starting from one-point tail bounds, we establish an upper tail large deviation principle for the directed landscape at the metric level. Metrics of finite rate are in one-to-one correspondence with measures supported on a set of countably many paths, and the rate function is given by a certain Kruzhkov entropy of these measures. As an application of our main result, we prove a large deviation principle for the directed geodesic.

math.PR↗
arXiv2023

Eigenvectors of the square grid plus GUE

Eigenvectors of the GUE-perturbed discrete torus with uniform boundary conditions retain some product structure for small perturbations but converge to discrete Gaussian waves for large perturbations. We determine where this phase transition happens.

math.PR↗
arXiv2023

Uniform convergence to the Airy line ensemble

We show that classical integrable models of last passage percolation and the related nonintersecting random walks converge uniformly on compact sets to the Airy line ensemble. Our core approach is to show convergence of nonintersecting Bernoulli random walks in all feasible directions in the parameter space. We then use coupling arguments to extend convergence to other models.

math.PR↗
arXiv2023

Large deviations for the interchange process on the interval and incompressible flows

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.

math.PR↗
arXiv2022

The scaling limit of the longest increasing subsequence

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.

math.PR↗
arXiv2021

RSK in last passage percolation: a unified approach

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.

math.PR↗
arXiv2021

Bulk properties of the Airy line ensemble

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.

math.PR↗
arXiv2020

Uniform point variance bounds in classical beta ensembles

In this paper, we give bounds on the variance of the number of points of the circular and the Gaussian $β$ ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on $β$.

math.PR↗
arXiv2020

The many faces of the stochastic zeta function

We introduce a framework to study the random entire function $ζ_β$ whose zeros are given by the Sine$_β$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power series representation built from Brownian motion. We study related distributions using stochastic differential equations. Our function is a uniform limit of characteristic polynomials in the circular beta ensemble; we give upper bounds on the rate of convergence. Most of our results are new even for classical values of $β$. We provide explicit moment formulas for $ζ$ and its variants, and we show that the Borodin-Strahov moment formulas hold for all $β$ both in the limit and for circular beta ensembles. We show a uniqueness theorem for $ζ$ in the Cartwright class, and deduce some product identities between conjugate values of $β$. The proofs rely on the structure of the Sine$_β$ operator to express $ζ$ in terms of a regularized determinant.

math.PR↗
arXiv2020

Entropy and expansion

Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edge-vertex inequality for factor-of-IID processes, Bowen's entropy inequalities, and Bollobás's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs.

math.PR↗
arXiv2019

The bead process for beta ensembles

The bead process introduced by Boutillier is a countable interlacing of the determinantal sine-kernel point processes. We construct the bead process for general 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.

math.PR↗
arXiv2019

The Local Limit of Random Sorting Networks

A sorting network is a geodesic path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of $an$ for $a\in[0,1]$ as $n\to\infty$. Here time is scaled by a factor of $1/n$ and space is not scaled. The limit is a swap process $U$ on $\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\sqrt{a(1-a)}$. To establish the existence of $U$, we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.

math.PR↗