Electrostatics of Coulomb gas, lattice paths, and discrete polynuclear growth

TL;DR

This paper analyzes the partition function of a Coulomb gas with external charges in a large-scale limit, connecting it to random matrix theory, lattice paths, and growth models, with explicit calculations and probabilistic estimates.

Contribution

It introduces a novel analysis of Coulomb gas partition functions in a double scaling limit and links these results to random matrix theory and discrete growth models.

Findings

Explicit formulas for the partition function in the scaling limit

Connections established between Coulomb gas and lattice path models

Probabilistic estimates for maximum height in growth models

Abstract

We study the partition function of a two-dimensional Coulomb gas on a circle, in the presence of external pointlike charges, in a double scaling limit where both the external charges and the number of gas particles are large. Our original motivation comes from studying amplitudes for multi-string emission from a decaying D-brane in the high energy limit. We analyze the scaling limit of the partition function and calculate explicit results. We also consider applications to random matrix theory. The partition functions can be related to random scattering, or to weights of lattice paths in certain growth models. In particular, we consider the discrete polynuclear growth model and use our results to compute the cumulative probability density for the height of long level-1 paths. We also obtain an estimate for an almost certain maximum height.