[Paper Review] Large-N asymptotic expansion for mean field models with Coulomb gas interaction

Gaëtan Borot, Alice Guionnet+1 more|arXiv (Cornell University)|Dec 23, 2013

Stochastic processes and statistical mechanics9 citations

TL;DR

This paper establishes a large-N asymptotic expansion for mean-field models with Coulomb gas interactions, generalizing β-ensembles to include arbitrary r-body interactions. Using complex and functional analysis, it proves the all-order asymptotic expansion is governed by a universal topological recursion, extending known results to non-pairwise interactions under analyticity and convexity conditions.

ABSTRACT

We derive the large-N, all order asymptotic expansion for a system of N particles with mean-field interactions on top of a Coulomb repulsion at temperature 1/β, under the assumptions that the interactions are analytic, off-critical, and satisfy a local strict convexity assumption.

Motivation & Objective

• To extend the large-N asymptotic analysis of β-ensembles to general mean-field models with r-body interactions beyond pairwise Coulomb repulsion.
• To establish conditions under which the full asymptotic expansion (up to O(N^{-∞})) is governed by a universal topological recursion.
• To generalize central limit theorems for linear statistics to multi-cut regimes with interference effects.
• To provide a rigorous framework for the all-order asymptotic expansion in models with analytic, off-critical interactions and local strict convexity.

Proposed method

• Formulates a mean-field particle system on a union of intervals with a joint measure including pairwise Coulomb repulsion and an r-body interaction scaled as N^{2−r}/r!.
• Assumes the r-body potential admits an asymptotic expansion in inverse powers of N, with coefficients independent of N.
• Applies tools from complex analysis and functional analysis to study the Fredholm integral operator associated with the equilibrium measure.
• Uses the principal value operator and singular integral equations to derive the equilibrium condition and invertibility of the associated operator.
• Establishes that the solution to the equilibrium condition leads to a unique solution via a bijective operator (id + N̲), ensuring well-posedness.
• Demonstrates that the full asymptotic expansion of the partition function and observables is determined by topological recursion, using the equilibrium density and covariance as initial data.

Experimental results

Research questions

• RQ1 Under what conditions does the all-order large-N asymptotic expansion exist for mean-field models with r-body interactions and Coulomb repulsion?
• RQ2 How does the topological recursion structure extend from β-ensembles to models with general r-body interactions?
• RQ3 What is the role of local strict convexity and analyticity in ensuring the convergence and uniqueness of the asymptotic expansion?
• RQ4 How do fluctuations of linear statistics behave in the multi-cut regime, and what interference effects emerge compared to the one-cut case?
• RQ5 Can the partition function's asymptotic expansion be fully determined by a universal recursion, even when interactions are not limited to pairwise terms?

Key findings

• The all-order large-N asymptotic expansion for the partition function is fully determined by a universal topological recursion, extending the known result for β-ensembles.
• The equilibrium measure is uniquely characterized by a Fredholm integral equation involving the principal value operator and a singular integral kernel.
• The operator governing the equilibrium condition is invertible, ensuring the existence and uniqueness of the solution under the stated analytic and convexity assumptions.
• The method allows for the derivation of a central limit theorem for linear statistics in the one-cut regime and its generalization to the multi-cut regime with interference effects.
• The asymptotic expansion of the r-body potential in inverse powers of N ensures that the interaction contributes at the same order as the 2-body Coulomb repulsion in the large-N limit.
• The solution to the equilibrium condition leads to a holomorphic function on the complex plane minus the support, satisfying jump conditions that allow reconstruction of the equilibrium density via singular integral equations.