Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble

@article{Nguyen2015NonintersectingBB,
  title={Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble},
  author={Gia Bao Nguyen and Daniel Remenik},
  journal={arXiv: Probability},
  year={2015}
}
We show that the squared maximal height of the top path among $N$ non-intersecting Brownian bridges starting and ending at the origin is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of K. Johansson's result that the supremum of the Airy$_2$ process minus a parabola has the Tracy-Widom GOE distribution, and as such it provides an explanation for how this distribution arises in models… 
View PDF on arXiv
15 Citations
Highly Influential Citations
2
Background Citations
6
Methods Citations
1
Results Citations
1

15 Citations

Citation Type

Citation Type

The hard-edge tacnode process for Brownian motion
We consider $N$ non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large $N$…
Constrained non-crossing Brownian motions, fermions and the Ferrari–Spohn distribution
A conditioned stochastic process can display a very different behavior from the unconditioned process. In particular, a conditioned process can exhibit non-Gaussian fluctuations even if the…
Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
We study the distribution of the supremum of the Airy process with $m$ wanderers minus a parabola, or equivalently the limit of the rescaled maximal height of a system of $N$ non-intersecting…
Brownian Bridges for Late Time Asymptotics of KPZ Fluctuations in Finite Volume
Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium…
Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions
This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random…
Non-crossing Brownian Paths and Dyson Brownian Motion Under a Moving Boundary
We compute analytically the probability $S(t)$ that a set of $N$ Brownian paths do not cross each other and stay below a moving boundary $g(\tau)= W \sqrt{\tau}$ up to time $t$. We show that for…
Extreme statistics of non-intersecting Brownian paths
We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and…
How flat is flat in random interface growth?
Domains of attraction are identified for the universality classes of one-point asymptotic fluctuations for the Kardar-Parisi-Zhang (KPZ) equation with general initial data. The criterion is based on…
The invariant measure of PushASEP with a wall and point-to-line last passage percolation
We consider an interacting particle system on the lattice involving pushing and blocking interactions, called PushASEP, in the presence of a wall at the origin. We show that the invariant measure of…
Scaling limits of permutations avoiding long decreasing sequences
We determine the scaling limit for permutations conditioned to have longest decreasing subsequence of length at most $d$. These permutations are also said to avoid the pattern $(d+1)d \cdots 2 1$ and…
...
...

References

SHOWING 1-10 OF 62 REFERENCES
Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials
We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to…
Nonintersecting Brownian Excursions
Author(s): Tracy, Craig A.; Widom, Harold | Abstract: We consider the process of $n$ Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and…
Non-intersecting Brownian walkers and Yang–Mills theory on the sphere
Exact distribution of the maximal height of p vicious walkers.
TLDR
The exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones and establishes a physical connection between vicious walkers and random matrix theory.
Non-colliding Brownian bridges and the asymmetric tacnode process
We consider non-colliding Brownian bridges starting from two points and returning to the same position. These positions are chosen such that, in the limit of large number of bridges, the two families…
Maximum distributions of bridges of noncolliding Brownian paths.
TLDR
The present work demonstrates that the extreme-value problems of noncolliding paths are related to random matrix theory, the representation theory of symmetry, and number theory.
Brownian Gibbs property for Airy line ensembles
Consider a collection of N Brownian bridges $B_{i}:[-N,N] \to \mathbb{R} $, Bi(−N)=Bi(N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak…
Critical behavior of nonintersecting Brownian motions at a tacnode
We study a model of n one‐dimensional, nonintersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the…
Non-colliding Brownian Motions and the Extended Tacnode Process
We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is…
Noncolliding Brownian Motion and Determinantal Processes
Abstract A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian…
...
...