Abstract
We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafa\"i, Hardy and Ma\"ida on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.
Highlights
We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold
This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function
Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function
Summary
We shall consider the model of a Coulomb gas on a Riemannian manifold introduced in [6, Subsection 4.1] and study its non-asymptotic behavior by obtaining a concentration inequality for the empirical measure around its limit. The metric we shall use on P(M ) is the function W1 : P(M ) × P(M ) → [0, ∞) defined by. In the two theorems above we have relaxed this inequality to r2 Pn (W1(in, π) ≥ r) ≤ exp −βn 4 + o(βn) and obtained a bound to o(βn) that does not depend on r. In this relaxed inequality and at a fixed r > 0 the order terms cannot be exact. The energy-distance comparison will be explained in Section 3 and it may be extended to include Green functions of some Laplace-type operators.
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