Amenability of linear-activity automaton groups (2009)
Amir, Gideon, Angel, Omer, Virag, Balint
We prove that every linear-activity automaton group is amenable. The proof is based on showing that a sufficiently symmetric random walk on a specially constructed degree 1 automaton group -- the...
The spectrum of the random environment and localization of noise (2008)
Cheliotis, Dimitrios, Virag, Balint
We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix...
Continuum limits of random matrices and the Brownian carousel (2007)
We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process....
Complex determinantal processes and H1 noise (2007)
Rider, Brian; University Of Colorado, Boulder; Brider@euclid.colorado.edu, Virag, Balint; University Of Toronto; Balint@math.toronto.edu
For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes with intensity ρ dν, where ν is the corresponding invariant measure. We show...
On the girth of random Cayley graphs (2007)
Gamburd, Alex, Hoory, Shlomo, Shahshahani, Mehrdad, Shalev, Aner, Virag, Balint
We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/2 and that random d-regular Cayley graphs of simple algebraic...
Random Sorting Networks (2006)
Angel, Omer, Holroyd, Alexander E., Romik, Dan, Virag, Balint
A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n->infinity the...
Complex determinantal processes and H1 noise (2006)
For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes with intensity rho dnu, where nu is the corresponding invariant measure. We show that as...
Beta ensembles, stochastic Airy spectrum, and a diffusion (2006)
Ramirez, Jose, Rider, Brian, Virag, Balint
We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x +...
The noise in the circular law and the Gaussian free field (2006)
Fill an n x n matrix with independent complex Gaussians of variance 1/n. As n approaches infinity, the eigenvalues {z_k} converge to a sum of an H^1-noise on the unit disk and an independent...
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process (2003)
Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint...
Amenability via random walks (2003)
Bartholdi, Laurent, Virag, Balint
We answer an open question of Grigorchuk and Zuk about amenability using random walks. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the...
We show that the past and future of half-plane Brownian motion at certain cutpoints are independent of each other after a conformal transformation. Like in Ito's excursion theory, the pieces between...
Dimension and randomness in groups acting on rooted trees (2002)
We explore the structure of the p-adic automorphism group Gamma of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We...
Random walks that avoid their past convex hull (2002)
Angel, Omer, Benjamini, Itai, Virag, Balint
We introduce planar random walk conditioned to avoid its past convex hull, and we show that it escapes at a positive limsup speed. Experimental results show that fluctuations from a limiting...
Anchored expansion and random walk (2001)
This paper studies anchored expansion, a non-uniform version of the strong isoperimetric inequality. We show that every graph with i-anchored expansion contains a subgraph with isoperimetric...
Fast graphs for the random walker (2001)
Consider the time T_oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T_oz in terms of the volume of z and the graph...
Random Walks on Finite Convex Sets of Lattice Points (1996)
This paper examines the convergence of nearest-neighbor random walks on convex subsets of the lattice Z d . The main result shows that for fixed d, O(fl 2 ) steps are sufficient for a walk to...