Subhroshekhar Ghosh

Rigidity and tolerance in point processes: Gaussian zeros 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.

Exact quantum algorithm to distinguish Boolean functions of different weights

In this work, we exploit the Grover operator for the weight analysis of a Boolean function, specifically to solve the weight-decision problem. The weight w is the fraction of all possible inputs for which the output is 1. The goal of the weight-decision problem is to find the exact weight w from the given two weights w1 and w2 satisfying a general weight condition as w1 + w2 = 1 and 0 <w 1 <w 2 < 1. First, we propose a limited weightdecision algorithm where the function has another constraint: a weight is in � w1 = sin 2 � k

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.

Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey

We present a brief survey of fluctuations and large deviations of particle systems with subextensive growth of the variance. These are called hyperuniform (or superhomogeneous) systems. We then discuss the relation between hyperuniformity and rigidity. In particular we give sufficient conditions for rigidity of such systems in d = 1, 2.

Palm measures and rigidity phenomena in point processes

We study the mutual regularity properties of Palm measures of point processes, and establish that a key determining factor for these properties is the rigidity-tolerance behaviour of the point process in question (for those processes that exhibit such behaviour). Thereby, we extend the results of Osada-Shirai, Bufetov and Olshanski to new ensembles, particularly those that are devoid of any determinantal structure. These include the zeroes of the standard planar Gaussian analytic function and several others.

Multivariate CLT follows from strong Rayleigh property.

Let

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

(X_1 , ldots , X_d)

Generalized Stealthy Hyperuniform Processes: Maximal Rigidity and the Bounded Holes Conjecture

be random variables taking nonnegative integer values and let

Determinantal processes and completeness of random exponentials: the critical case

f(z_1, ldots , z_d)

Large Deviations for Zeros of Random Polynomials with i.i.d. Exponential Coefficients

be the probability generating function. Suppose that