Abstract
We examine a Coulomb gas consisting of $n$ identical repelling point charges at an arbitrary inverse temperature $\beta$, subjected to a suitable external field. We prove that the gas is effectively localized to a small neighbourhood of the "droplet" -- the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to $$\sqrt{\dfrac {\log n}{\beta n}}.$$ This order of magnitude is known to be "tight" when $\beta=1$ and the external field is radially symmetric. In addition, we prove estimates for the one-point function in a neighbourhood of the droplet, proving in particular a fast uniform decay as one moves beyond a distance roughly of the order $\sqrt{\frac {\log n}{\beta n}}$ from the droplet.
Highlights
Introduction and main resultsThe planar Coulomb gas is a random configuration consisting of many identical repelling point charges {ζi}n1 in C
F = q · e−nQ/2 where q is a holomorphic polynomial of degree at most n − 1
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Summary
The planar Coulomb gas is a random configuration consisting of many (but finitely many) identical repelling point charges {ζi}n1 in C. The function Q, which is called an external potential, is fairly general but not quite arbitrary; precise assumptions are given below. We define a positive constant c0 = c0[Q] by c0 := min{∆Q(η); η ∈ ∂S} Given these preliminaries, we have the following theorem. It could be said that Theorem 2 gives more detailed information about exactly how “localized” the gas is about the droplet To illustrate this point, we may observe that if we fix a β > 0 and choose t = tn so that tn → ∞ and tn/ log n → 0 as n → ∞, (1.11) implies. As is well-known, we can interpret the Coulomb gas {ζj }nj=1 in external potential Q at inverse temperature β = 1 as eigenvalues of normal random matrices.
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