• Corpus ID: 221738954

Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution

@article{Liechty2020AiryPW,
  title={Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution},
  author={Karl Liechty and Gia Bao Nguyen and Daniel Remenik},
  journal={arXiv: Probability},
  year={2020}
}
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 as $N\to\infty$, where the first $N-m$ paths start and end at the origin and the remaining $m$ go between arbitrary positions. The distribution provides a $2m$-parameter deformation of the Tracy--Widom GOE distribution, which is recovered in the limit corresponding to all Brownian paths… 
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References

SHOWING 1-10 OF 100 REFERENCES
Airy processes with wanderers and new universality classes.
Consider n + m nonintersecting Brownian bridges, with n of them leaving from 0 at time t = -1 and returning to 0 at time t = -1, while the m remaining ones (wanderers) go from m points a(i) to m…
Local behavior and hitting probabilities of the Airy1 process
We obtain a formula for the $n$-dimensional distributions of the Airy$_1$ process in terms of a Fredholm determinant on $L^2(\rr)$, as opposed to the standard formula which involves extended kernels,…
Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces
We compute the joint probability density function (jpdf) PN(M,τM) of the maximum M and its position τM for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For…
The KPZ fixed point
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process with arbitrary initial condition. The…
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…
Supremum of the Airy2 Process Minus a Parabola on a Half Line
AbstractLet $\mathcal {A}_{2}(t)$ be the Airy2 process. We show that the random variable $$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$ has the same distribution as…
Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
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…
The k-tacnode process
The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two…
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…
One-sided reflected Brownian motions and the KPZ fixed point
Abstract We consider the system of one-sided reflected Brownian motions that is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities,…
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