Showing papers by "Bálint Virág published in 2007"
TL;DR: In this article, it was shown that for a uniform random sorting network, as n → ∞ the space-time process of swaps converges to the product of semicircle law and Lebesgue measure.
98 citations
TL;DR: It is proved 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 thatrandom d- regular Cayley graph of simple algebraic groups over F_q asymPToticallyalmost surely haveGirth at at least log_{d -1} |G|/dim(G).
Abstract: 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 groups over F_q asymptotically almost surely have girth at least log_{d-1}|G|/dim(G). For the symmetric p-groups the girth is between log log |G| and (log|G|)^alpha with alpha<1. Several conjectures and open questions are presented.
34 citations
TL;DR: For the plane, sphere, and hyperbolic plane, the canonical invariant determinantal point processes were studied in this article, where it was shown that these processes converge to invariant noise.
Abstract: For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes $\mathcal Z_\rho$ with intensity $\rho d
u$, where $
u$ is the corresponding invariant measure. We show that as $\rho\to\infty$, after centering, these processes converge to invariant $H^1$ noise. More precisely, for all functions $f\in H^1(
u) \cap L^1(
u)$ the distribution of $\sum_{z\in \mathcal Z} f(z)-\frac{\rho}{\pi} \int f d
u$ converges to Gaussian with mean zero and variance $ \frac{1}{4 \pi} \|f\|_{H^1}^2$.
30 citations
TL;DR: In this article, it was shown 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.
Abstract: 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. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.
The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $\beta$, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta=2.
2 citations