[Paper Review] 2D Coulomb Gases and the Renormalized Energy

Étienne Sandier, Sylvia Serfaty|arXiv (Cornell University)|Jan 17, 2012

Random Matrices and Applications44 references19 citations

TL;DR

This paper analyzes 2D Coulomb gases with general potential and inverse temperature β, linking macroscopic equilibrium measures to microscopic point configurations via the renormalized energy W. It establishes a next-order asymptotic expansion of the partition function, fluctuation estimates at the microscale, and a large deviations principle showing that high-β systems crystallize toward W-minimizers—conjectured to be Abrikosov triangular lattices in the β→∞ limit.

ABSTRACT

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary <i>β</i> the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case <i>β</i> = ∞ corresponds to "weighted Fekete sets" and also falls within our analysis.<br> It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni [BZ].<br> By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" <i>W</i>, a Coulombian interaction for points in the plane introduced in [SS1],which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of <i>W</i> have exponentially small probability. When <i>β</i> → ∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of <i>W</i>. In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of <i>W</i>, which are conjectured to be "Abrikosov" triangular lattices.

Motivation & Objective

• To understand the microscopic structure of 2D Coulomb gases at finite and infinite β.
• To connect the equilibrium measure to the renormalized energy W, a measure of point configuration disorder.
• To derive asymptotic expansions and large deviations for fluctuations around equilibrium at the microscale.
• To show that in the β→∞ limit, configurations converge to W-minimizers, corresponding to weighted Fekete sets.

Proposed method

• Decompose the Hamiltonian into equilibrium and fluctuation parts to isolate the role of the renormalized energy W.
• Use the renormalized energy W as a microscopic interaction measure for infinite point configurations in the plane.
• Apply large deviations principles to quantify the probability of rare configurations with high W values.
• Derive a next-order asymptotic expansion of the partition function using W as a key component.
• Analyze the β→∞ limit to show that the system must minimize W, implying crystallization.
• Leverage known results on W-minimizers to infer that weighted Fekete sets converge to Abrikosov lattices at the microscale.

Experimental results

Research questions

• RQ1 How does the renormalized energy W govern the microscopic structure of 2D Coulomb gases?
• RQ2 What is the asymptotic behavior of the partition function beyond the leading-order equilibrium measure?
• RQ3 What is the probability of observing point configurations with high W values at the microscale?
• RQ4 How does the system behave as β→∞, and does it crystallize toward W-minimizers?
• RQ5 Do weighted Fekete sets asymptotically resemble Abrikosov triangular lattices at the microscale?

Key findings

• The partition function admits a next-order asymptotic expansion involving the renormalized energy W.
• Fluctuations from the equilibrium measure at the microscale are quantitatively controlled by W.
• Configurations with W above a certain threshold have exponentially small probability, establishing a large deviations principle.
• As β→∞, the system is forced to minimize W, implying that weighted Fekete sets must microscopically resemble W-minimizers.
• The minimizers of W are conjectured to be Abrikosov triangular lattices, suggesting that weighted Fekete sets converge to such structures at the microscale.
• The renormalized energy W provides an effective microscopic description of point disorder in 2D Coulomb systems.