# Some connections between the Classical Calogero-Moser model and the Log Gas

## TL;DR

This paper explores the deep mathematical connections between the Calogero-Moser particle system and the log-gas model from random matrix theory, revealing shared equilibrium configurations and related oscillation properties through analytical and numerical methods.

## Contribution

It uncovers the relationship between the Hessians of the two models and compares their finite temperature properties using Monte Carlo simulations.

## Key findings

- Shared minimum energy configuration with Hermite polynomial zeros
- Hessian of Calogero-Moser is the square of the log-gas Hessian
- Numerical results agree with theoretical predictions for both models

## Abstract

In this work we discuss connections between a one-dimensional system of $N$ particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero-Moser model) and the log-gas model which appears in random matrix theory. Both models have the same minimum energy configuration, with the particle positions given by the zeros of the Hermite polynomial. Moreover, the Hessian describing small oscillations around equilibrium are also related for the two models. The Hessian matrix of the Calogero-Moser model is the square of that of the log-gas. We explore this connection further by studying finite temperature equilibrium properties of the two models through Monte-Carlo simulations. In particular, we study the single particle distribution and the marginal distribution of the boundary particle which, for the log-gas, are respectively given by the Wigner semi-circle and the Tracy-Widom distribution. For particles in the bulk, where typical fluctuations are Gaussian, we find that numerical results obtained from small oscillation theory are in very good agreement with the Monte-Carlo simulation results for both the models. For the log-gas, our findings agree with rigorous results from random matrix theory.