Related papers: The generating function for the Airy point process…
We study the Tracy-Widom (TW) distribution $f_\beta(a)$ in the limit of large Dyson index $\beta \to +\infty$. This distribution describes the fluctuations of the rescaled largest eigenvalue $a_1$ of the Gaussian (alias Hermite) ensemble…
We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near…
We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlev\'{e} III equation. This Fredholm determinant describes the…
$\tau$-functions of certain Painlev\'e equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. This paper is a step…
The distribution function of the free energy fluctuations in one-dimensional directed polymers with $\delta$-correlated random potential is studied by mapping the replicated problem to a many body quantum boson system with attractive…
In a recent contribution, Dotsenko establishes a Fredholm determinant formula for the two-point distribution of the KPZ equation in the long time limit and starting from narrow wedge initial conditions. We establish that his expression is…
We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities…
Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy_1-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest…
We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels \`a la Its-Izergin-Korepin-Slavnov (IIKS) and hence related to suitable Riemann-Hilbert problems,…
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, on $L^2(\{1,...,n\}\times\rr)$. The…
In random matrix theory (RMT), the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian…
We establish the relation between two objects: an integrable system related to Painlev\'e II equation, and the symplectic invariants of a certain plane curve S(TW). This curve describes the average eigenvalue density of a random hermitian…
We study the joint probability generating function for $k$ occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of…
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…
We prove a Tracy-Widom type formula for the generating function of occupancy numbers on several disjoint intervals of the higher order Airy point processes. The formula is related to a new vector-valued Painlev\'e II hierarchy we define,…
Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in…
Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values…
The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a…
The purpose of this article is to develop a theory behind the occurrence of "path-integral" kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants…
We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve \Sigma_{TW} describing the average eigenvalue density of a random hermitian matrix…