Related papers: Nonlinear PDEs for gap probabilities in random mat…
The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more…
We find the probability of two gaps of the form $(sc,sb)\cup (sa,+\infty)$, $c<b<a<0$, for large $s>0$, in the edge scaling limit of the Gaussian Unitary Ensemble of random matrices, including the multiplicative constant in the asymptotics.
As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to…
We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We…
For a broad class of point processes, including determinantal point processes, we construct associated marked and conditional ensembles, which allow to study a random configuration in the point process, based on information about a randomly…
In these lecture notes we present some connections between random matrices, the asymmetric exclusion process, random tilings. These three apparently unrelated objects have (sometimes) a similar mathematical structure, an interlacing…
Building on our previous work on Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to inverse problems for linear and nonlinear elliptic partial differential equations. The proposed scheme…
In a recent study of large non-null sample covariance matrices, a new sequence of functions generalizing the GUE Tracy-Widom distribution of random matrix theory was obtained. This paper derives Painlev\'e formulas of these functions and…
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian…
We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions. This is achieved by exploiting the connection between these…
In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices the probability that an interval of length $s$ contains no eigenvalues is the Fredholm determinant of the sine kernel $\sin(x-y)\over\pi(x-y)$ over this…
We consider a family of determinantal random point processes on the two-dimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures…
We consider two versions of discrete time totally asymmetric simple exclusion processes (TASEPs) with geometric and Bernoulli random hopping probabilities. For the process mixed with these and continuous time dynamics, we obtain a single…
In this paper, we consider the Wright's generalized Bessel kernel $K^{(\alpha,\theta)}(x,y)$ defined by $$\theta x^{\alpha}\int_0^1J_{\frac{\alpha+1}{\theta},\frac{1}{\theta}}(ux)J_{\alpha+1,\theta}((uy)^{\theta})u^\alpha\,\mathrm{d} u,…
We consider the squared singular values of the product of $M$ standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are…
We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support.…
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 $m$ points $b_i$. First, we keep $m$…
We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are…
The explicit expression for the two time free energy distribution function in one-dimensional random directed polymers is derived in terms of the Bethe ansatz replica technique. It is show that such type of the distribution function can be…
We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma):=\det(I-\gamma K_s\upharpoonright_{L^2(-1,1)})$, $0\leq\gamma\leq 1$. The operator $K_s$…