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Subhro Ghosh

Publications and source records attributed to Subhro Ghosh.

23 records · Page 2Linked to original sources
arXiv2015

Rigidity and Tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues

Let X be a translation invariant point process on the complex plane and let D be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration obtained by taking the points of X outside D tell us about the point configuration inside D? We show that for the Ginibre ensemble, it determines the number of points in D. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that it determines the number as well as the centre of mass of the points in D. Further, in both models we prove that the outside says "nothing more" about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold.

math.PR↗
arXiv2014

Determinantal processes and completeness of random exponentials: the critical case

For a locally finite point set $Λ\subset \mathbb{R}$, consider the collection of exponential functions given by $\mathcal{E}_Λ:= \{e^{i λx} : λ\in L \}$. We examine the question whether $\mathcal{E}_Λ$ spans the Hilbert space $L^2[-π,π]$, when $Λ$ is random. For several point processes of interest, this belongs to a certain critical case of the corresponding question for deterministic $Λ$, about which little is known. For $Λ$ the continuum sine kernel process, obtained as the bulk limit of GUE eigenvalues, we establish that $\mathcal{E}_Λ$ is indeed complete. We also answer an analogous question on $\mathbb{C}$ for the Ginibre ensemble, arising as weak limits of certain non-Hermitian random matrix eigenvalues. In fact we establish completeness for any "rigid" determinantal point process in a general setting. In addition, we partially answer two questions due to Lyons and Steif about stationary determinantal processes on $\mathbb{Z}^d$.

math.PR↗
arXiv2012

Rigidity and Tolerance in Gaussian zeroes and Ginibre eigenvalues: quantitative estimates

Let $Π$ be a translation invariant point process on the complex plane $\C$ and let $\D \subset \C$ be a bounded open set whose boundary has zero Lebesgue measure. We study the conditional distribution of the points of $Π$ inside $\D$ given the points outside $\D$. When $Π$ is the Ginibre ensemble or the Gaussian zero process, it been shown in \cite{GP} that this conditional distribution is mutually absolutely continuous with the Lebesgue measure on its support. In this paper, we refine the result in \cite{GP} to show that the conditional density is, roughly speaking, comparable to a squared Vandermonde density. In particular, this shows that even under spatial conditioning, the points exhibit repulsion which is quadratic in their mutual separation.

math.PR↗
arXiv2012

Continuum Percolation for Gaussian zeroes and Ginibre eigenvalues

We study continuum percolation on certain negatively dependent point processes on \R^2. Specifically, we study the Ginibre ensemble and the planar Gaussian zero process, which are the two main natural models of translation invariant point processes on the plane exhibiting local repulsion. For the Ginibre ensemble, we establish the uniqueness of infinite cluster in the supercritical phase. For the Gaussian zero process, we establish that a non-trivial critical radius exists, and we prove the uniqueness of infinite cluster in the supercritical regime.

math.PR↗