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Bálint Virág

Publications and source records attributed to Bálint Virág.

At least 37 records · Page 2Linked to original sources
arXiv2018

Circular support in random sorting networks

A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 2 1$ in the Cayley graph of the symmetric group generated by adjacent transpositions. For a uniform random sorting network, we prove that in the global limit, particle trajectories are supported on $π$-Lipschitz paths. We show that the weak limit of the permutation matrix of a random sorting network at any fixed time is supported within a particular ellipse. This is conjectured to be an optimal bound on the support. We also show that in the global limit, trajectories of particles that start within distance $ε$ of the edge are within $\sqrt{2ε}$ of a sine curve in uniform norm.

math.PR↗
arXiv2018

Tracy-Widom fluctuations in 2D random Schrodinger operators

We construct a random Schrodinger operator on a subset of the hexagonal lattice and study its smallest positive eigenvalues. Using an asymptotic mapping, we relate them to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the random Schrodinger operator. We also relate the distribution of its k smallest positive eigenvalues to the nonintersecting partition functions of order k.

math.PR↗
arXiv2017

Operator limit of the circular beta ensemble

We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate on the beta-dependence in the $\text{Sine}_β$ process.

math.PR↗
arXiv2016

Limits of spiked random matrices II

The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and Péché [Duke Math. J. (2006) 133 205-235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schrödinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($β=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson's Brownian motion, or alternatively a linear parabolic PDE; here $β$ appears simply as a parameter. At $β=2$, the PDE appears to reconcile with known Painlevé formulas for these $r$-parameter deformations of the GUE Tracy-Widom law.

math.PR↗
arXiv2016

Spectral measures of factor of i.i.d. processes on vertex-transitive graphs

We prove that a measure on $[-d, d]$ is the spectral measure of a factor of i.i.d. process on a vertex-transitive infinite graph if and only if it is absolutely continuous with respect to the spectral measure of the graph. Moreover, we show that the set of spectral measures of factor of i.i.d. processes and that of $\bar d_2$-limits of factor of i.i.d. processes are the same.

math.PR↗
arXiv2016

The Sine$_β$ operator

We show that Sine$_β$, the bulk limit of the Gaussian $β$-ensembles is the spectrum of a self-adjoint random differential operator \[ f\to 2 {R_t^{-1}} \left[ \begin{array}{cc} 0 &-\tfrac{d}{dt} \tfrac{d}{dt} &0 \end{array} \right] f, \qquad f:[0,1)\to \mathbb R^2, \] where $R_t$ is the positive definite matrix representation of hyperbolic Brownian motion with variance $4/β$ in logarithmic time. The result connects the Montgomery-Dyson conjecture about the Sine$_2$ process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Pólya conjecture and de Brange's attempt to prove the Riemann hypothesis. We identify the Brownian carousel as the Sturm-Liouville phase function of this operator. We provide similar operator representations for several other finite dimensional random ensembles and their limits: finite unitary or orthogonal ensembles, Hua-Pickrell ensembles and their limits, hard-edge $β$-ensembles, as well as the Schrödinger point process. In this more general setting, hyperbolic Brownian motion is replaced by a random walk or Brownian motion on the affine group. Our approach provides a unified framework to study $β$-ensembles that has so far been missing in the literature. In particular, we connect Itô's classification of affine Brownian motions with the classification of limits of random matrix ensembles.

math.PR↗
arXiv2016

Dyson's spike for random Schroedinger operators and Novikov-Shubin invariants of groups

We study Schroedinger operators with random edge weights and their expected spectral measures $μ_H$ near zero. We prove that the measure exhibits a spike of the form $μ_H(-\varepsilon,\varepsilon) \sim \frac{C}{| \log\varepsilon|^2}$ (first observed by Dyson), without assuming independence or any regularity of edge weights. We also identify the limiting local eigenvalue distribution, which is different from Poisson and the usual random matrix statistics. We then use the result to compute Novikov-Shubin invariants for various groups, including lamplighter groups and lattices in the Lie group Sol.

math.PR↗
arXiv2015

Independence ratio and random eigenvectors in transitive graphs

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $λ_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least \[q=\frac{1}{2}-\frac{3}{4π}\arccos\biggl(\frac{1-λ_{\min}}{4}\biggr).\] The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least $q-o(1)$. We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.

math.PR↗
arXiv2015

A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix $T_0$. Focusing on the eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required $T_0$ to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some result by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular we solve a problem posed in their work.

math-ph↗
arXiv2015

The Liouville property for groups acting on rooted trees

We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure.

math.GR↗
arXiv2015

Non-Liouville groups with return probability exponent at most 1/2

We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \gtrsim e^{-n^{1/2 + o(1)}}$. Recent results suggest that $1/2$ is indeed the smallest possible return probability exponent for non-Liouville groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.

math.GR↗
arXiv2013

Patterns in Sinai's walk

Sinai's random walk in random environment shows interesting patterns on the exponential time scale. We characterize the patterns that appear on infinitely many time scales after appropriate rescaling (a functional law of iterated logarithm). The curious rate function captures the difference between one-sided and two-sided behavior.

math.PR↗
arXiv2013

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that $|λ| \leq 2\sqrt{d-1}$. We then use these approximations for $λ= -2\sqrt{d-1}$ to produce factor of i.i.d. independent sets on regular trees.

math.CO↗
arXiv2012

Random walks veering left

We study coupled random walks in the plane such that, at each step, the walks change direction by a uniform random angle plus an extra deterministic angle θ. We compute the Hausdorff dimension of the θfor which the walk has an unusual behavior. This model is related to a study of the spectral measure of some random matrices.

math.PR↗
arXiv2011

The Ginibre ensemble and Gaussian analytic functions

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Pólya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

math.PR↗
arXiv2011

Absolute continuity of the limiting eigenvalue distribution of the random Toeplitz matrix

We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schrödinger operators. The similar question for Hankel matrices remains open.

math.PR↗
arXiv2011

Limits of spiked random matrices I

Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank-one spiked real Wishart setting and its general beta analogue, proving a conjecture of Baik, Ben Arous and Péché (2005). We also treat shifted mean Gaussian orthogonal and beta ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which beta appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at beta=2,4, yielding in particular a new and simple proof of the Painlevé representations for these Tracy-Widom distributions.

math.PR↗