Hole Probabilities for Finite and Infinite Ginibre Ensembles

Adhikari, Kartick and Reddy, Nanda Kishore (2017) Hole Probabilities for Finite and Infinite Ginibre Ensembles. In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES (21). pp. 6694-6730.

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Abstract

We study the hole probabilities of the infinite Ginibre chi(infinity) ensemble a determinantal point process on the complex plane with the kernel K(z, w) = 1/pi e(z (w) over bar -1/2 vertical bar w vertical bar 2) with respect to the Lebesgue measure on the complex plane. Let U be an open subset of open unit disk ID and chi(infinity)(rU) denote the number of points of chi(infinity), that fall in rU. Then, under some conditions on U, we show that lim(r ->infinity) 1/r(4) log Pchi(infinity) (rU) = 0] = R-empty set - R-U, where empty set is the empty set and Ru := inf(mu is an element of P(UC)) {integral integral log 1/vertical bar z-w vertical bar d mu(z)d mu(w) + integral vertical bar z vertical bar(2) d mu(z)}, P(U-C) is the space of all compactly supported probability measures with support in U-C. Using potential theory, we give an explicit formula for Ru, the minimum possible energy of a probability measure compactly supported on U-c under logarithmic potential with a quadratic external field. Moreover, we calculate Ru explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle, and half disk.

Item Type:	Journal Article
Publication:	INTERNATIONAL MATHEMATICS RESEARCH NOTICES
Additional Information:	Copy right for this article belongs to the OXFORD UNIV PRESS, GREAT CLARENDON ST, OXFORD OX2 6DP, ENGLAND
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	24 Nov 2017 10:12
Last Modified:	24 Nov 2017 10:12
URI:	http://eprints.iisc.ac.in/id/eprint/58302

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