Large deviations for zeros of $P(φ)_2$ random polynomials

Details Large deviations for zeros of $P(φ)_2$ random polynomials Large deviations for zeros of $P(φ)_2$ random polynomials

2010-09-27

arxiv.org/abs/1009.5142v1

J. Stat. Phys. 143 (2011), no. 4, 619-635

Mathematics

Probability

This is a continuation of the article Large deviations of empirical zero point measures on Riemann surfaces, I: $g = 0$ by O.

Scientific paper

We extend results of Zeitouni-Zelditch on large deviations principles for zeros of Gaussian random polynomials $s$ in one complex variable to certain non-Gaussian ensembles that we call $P(\phi)_2$ random polynomials. The probability measures are of the form $e^{- S(f)} df$ where the actions $S(f)$ are finite dimensional analgoues of those of $P(\phi)_2$ quantum field theory. The speed and rate function are the same as in the associated Gaussian case. As a corollary, we prove that the expected distribution of zeros in the $P(\phi)_2$ ensembles tends to the same equilibrium measure as in the Gaussian case.

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