For high-temperature 1D log gases, the rescaled largest particle obeys a large deviation principle with the iid rate function x^d - 1, and tridiagonal matrices with Gaussian tails obey the analogous principle with rate x^2 - 1.
Emergence of a Poisson process in weakly interacting particle systems
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abstract
We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse temperature $\beta$ satisfies $N^{-1} \ll \beta \ll N^{-\frac{1}{2}}$, where $N$ is the number of particles. This expands the temperature regime for which convergence to a Poisson point process has been proved.
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Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature
For high-temperature 1D log gases, the rescaled largest particle obeys a large deviation principle with the iid rate function x^d - 1, and tridiagonal matrices with Gaussian tails obey the analogous principle with rate x^2 - 1.