Results 1  10
of
12
Brownian gibbs property for airy line ensembles
"... 1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is ad ..."
Abstract

Cited by 30 (8 self)
 Add to MetaCart
(Show Context)
1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is added to each of the curves of this scaling limit, an xtranslation invariant process sometimes called the multiline Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and nonintersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers ” and “outliers”. We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property – called the Brownian Gibbs property – of being invariant under the following action. Select an index 1 ≤ k ≤ N and erase Bk on a fixed time interval (a, b) ⊆ (−N, N); then replace this erased curve with a new curve on (a, b) according to the law of a Brownian bridge between the two existing endpoints ( a, Bk(a) ) and ( b, Bk(b) ) , conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edgescaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the longstanding conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multiline Airy process, and readily yields several other interesting properties of this process. 1.
Sparse regular random graphs: Spectral density and eigenvectors
"... Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized. 1.
Ordered Random Walks
, 2006
"... We construct the conditional version of k independent and identically distributed random walks on R given that they stay in strict order at all times. This is a generalisation of socalled noncolliding or nonintersecting random walks, the discrete variant of Dyson’s Brownian motions, which have b ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
(Show Context)
We construct the conditional version of k independent and identically distributed random walks on R given that they stay in strict order at all times. This is a generalisation of socalled noncolliding or nonintersecting random walks, the discrete variant of Dyson’s Brownian motions, which have been considered yet only for nearestneighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob htransform with some positive regular function V that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson’s Brownian motions, the continuous analogue.
Conditional limit theorems for ordered random walks
"... i E l e c t r o n J o u r n a l o f ..."
(Show Context)
Random walks in cones
, 2011
"... Abstract. We study the asymptotic behaviour of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We study the asymptotic behaviour of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by the Brownian motion. For the proof of local limit theorems we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results: ordered random walks and lattice path enumeration.
Nonintersecting Brownian motions on the unit circle
, 2014
"... We consider an ensemble of n nonintersecting Brownian particles on the unit circle with diffusion parameter n−1/2, which are conditioned to begin at the same point and to return to that point after time T, but otherwise not to intersect. There is a critical value of T which separates the subcritical ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We consider an ensemble of n nonintersecting Brownian particles on the unit circle with diffusion parameter n−1/2, which are conditioned to begin at the same point and to return to that point after time T, but otherwise not to intersect. There is a critical value of T which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n → ∞, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painleve ́ II equation of size 2 × 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.