[Paper Review] Finite-particle approximations for interacting Brownian particles with logarithmic potentials

Yosuke Kawamoto, Hirofumi Osada|arXiv (Cornell University)|Jul 23, 2016

Random Matrices and Applications15 references3 citations

TL;DR

This paper establishes finite-particle approximations for infinite systems of interacting Brownian motions with logarithmic potentials, proving convergence to infinite-dimensional SDEs (ISDEs) via two general theorems on tightness and uniqueness. The key contribution is a rigorous framework for deriving ISDEs from finite-particle systems, validated for Airy β and Ginibre processes with long-range logarithmic interactions.

ABSTRACT

We prove the convergence of $ N $-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). % For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. % In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. % In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. % We apply the second general theorem to Airy$ _{β}$ interacting Brownian motion with $ β= 1,2,4$, and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.

Motivation & Objective

• To rigorously derive infinite-dimensional stochastic differential equations (ISDEs) from finite-particle systems of interacting Brownian motions with logarithmic potentials.
• To address the challenge of long-range, unbounded logarithmic interactions, which make the limit ISDEs non-trivial to detect from finite-dimensional SDEs.
• To establish general sufficient conditions for tightness and convergence of finite-particle solutions to solutions of ISDEs, applicable to both reversible and non-reversible systems.
• To validate the general theorems on specific models: Airy β interacting Brownian motions (β=1,2,4) and Ginibre interacting Brownian motion in two dimensions.
• To extend the framework to grand canonical Gibbs measures with Ruelle-class potentials, including Lennard-Jones 6-12 and Riesz potentials.

Proposed method

• Proposes a first general theorem on tightness of solutions to finite-particle SDEs, proving that limit points solve the corresponding ISDE under a sufficient condition, with convergence if uniqueness holds.
• Introduces a second, simplified theorem for reversible systems by reducing the tightness condition, derived from the first theorem.
• Applies the second theorem to Airy β and Ginibre processes by verifying the required conditions on correlation functions and logarithmic derivatives.
• Uses the DLR framework and regular conditional probabilities to define Gibbs measures and ensure consistency of finite-particle approximations.
• Employs Itô's formula and integration by parts to derive the SDEs for finite and infinite systems, particularly for logarithmic and Riesz-type potentials.
• Establishes uniform bounds on correlation functions (e.g., ρ_gin^{N,1} ≤ 1/π) to verify the tightness condition (J2)–(J5) and the logarithmic derivative condition (J4)–(J5).

Experimental results

Research questions

• RQ1 Can finite-particle systems of interacting Brownian motions with logarithmic potentials converge to an infinite-dimensional SDE (ISDE) in the thermodynamic limit?
• RQ2 What general sufficient conditions ensure tightness of finite-particle SDE solutions and convergence to an ISDE, especially when interactions are long-range and unbounded?
• RQ3 How can the framework be adapted to non-reversible systems, and what simplifications arise in the reversible case?
• RQ4 To what extent do the results extend to other Ruelle-class potentials, such as Lennard-Jones 6-12 and Riesz potentials?
• RQ5 Can the logarithmic derivative condition and correlation function bounds be verified for specific point processes like Ginibre and Airy β?

Key findings

• The first general theorem establishes sufficient conditions for tightness of finite-particle SDE solutions, ensuring that limit points solve the corresponding ISDE, with full convergence if solution uniqueness holds.
• The second theorem simplifies the tightness condition for reversible systems, enabling straightforward application to Airy β and Ginibre processes.
• For the Ginibre interacting Brownian motion in two dimensions, the limit ISDE is derived with a logarithmic interaction potential Ψ(x) = -log|x|, and the logarithmic derivative condition is verified using the uniform bound ρ_gin^{N,1} ≤ 1/π.
• The Airy β process (β=1,2,4) arises as the soft-edge limit of Gaussian ensembles, and its ISDE is rigorously derived via the second theorem.
• The framework applies to grand canonical Gibbs measures with Ruelle-class potentials, including Lennard-Jones 6-12 and Riesz potentials, provided correlation functions are uniformly bounded.
• For Riesz potentials with Ψ_a(x) = β/a |x|^{-a} (d < a ∈ ℕ), the ISDE is shown to be dX_t^i = dB_t^i + (β/2) ∑_{j≠i} (X_t^i - X_t^j)/|X_t^i - X_t^j|^{2+a} dt, with convergence established under uniform correlation bounds.