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Register a new userDeterminantal point processes from symplectic and orthogonal characters and applications
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Dan Betea
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We show that the symplectic and orthogonal character analogues of Okounkovs Schur measure (on integer partitions) are determinantal, with explicit correlation kernels. We apply this to prove certain Borodin-Okounkov-Gessel-type results concerning Toeplitz+Hankel and Fredholm determinants; a SzegH{o}-type limit theorem; an edge Baik-Deift-Johansson-type asymptotical result for certain symplectic and orthogonal analogues of the poissonized Plancherel measure; and a similar result for actual poissonized Plancherel measures supported on almost symmetric partitions.
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The gamma kernels are a family of projection kernels $K^{(z,z)}=K^{(z,z)}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Eulers gamma function and depend on two continuous parameters $z,z$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z)}$ serves as a correlation kernel for a determinantal measure $M^{(z,z)}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ ldots, K^{(z-1,z-1)}, ; K^{(z,z)},; K^{(z+1,z+1)}, ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z)$, the kernel $K^{(z,z)}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z+1)}$, and the one-point Palm distributions for the measure $M^{(z,z)}$ are absolutely continuous with respect to $M^{(z+1,z+1)}$. We also explicitly compute the corresponding Radon-Nikodym derivatives and show that they are given by certain normalized multiplicative functionals.
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, omega)$ over a domain $U subset mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^infty(U)$-module structure of $A^2(U, omega)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
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We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow band about the real line, of height proportional to $tfrac 1 N$, where $N$ denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Wards equation and a property which we call cross-section convergence, which relates the large-$N$ limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko-Pastur law for LUE.
Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution $q$ over the subsets of a ground set, we aim to distinguish whether $q$ is a DPP distribution, or $epsilon$-far from all DPP distributions in $ell_1$-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions.
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