[Paper Review] The Large-$N$ Limits of Brownian Motions on $\mathbb{GL}_N$

Todd Kemp|arXiv (Cornell University)|Jun 25, 2013

Random Matrices and Applications12 references3 citations

TL;DR

This paper introduces a two-parameter family of Brownian motions on the general linear group GL_N, driven by a family of invariant Riemannian metrics parameterized by r, s > 0. It proves that as N → ∞, the noncommutative distribution of these matrix-valued processes converges to a free Itô process b_{r,s}(t), solving a free stochastic differential equation. The result resolves Biane's open problem on the large-N limit of GL_N Brownian motion, unifying free unitary and multiplicative Brownian motions as special cases.

ABSTRACT

We introduce a two-parameter family of diffusion processes $(B_{r,s}^N(t))_{t\ge 0}$, $r,s>0$, on the general linear group $\mathbb{GL}_N$ that are Brownian motions with respect to certain natural metrics on the group. At the same time, we introduce a two-parameter family of free Itô processes $(b_{r,s}(t))_{t\ge 0}$ in a faithful, tracial $W^\ast$-probability space, and we prove that the full process $(B^N_{r,s}(t))_{t\ge 0}$ converges to $(b_{r,s}(t))_{t\ge 0}$ in noncommutative distribution as $N o\infty$ for each $r,s>0$. The processes $(b_{r,s}(t))_{t\ge 0}$ interpolate between the free unitary Brownian motion when $(r,s)=(1,0)$, and the free multiplicative Brownian motion when $r=s=\frac12$; we thus resolve the open problem of convergence of the Brownian motion on $\mathbb{GL}_N$ posed by Biane in 1997.

Motivation & Objective

• To define a family of Brownian motions on GL_N using r,s-parameterized invariant Riemannian metrics.
• To establish the large-N limit of these matrix-valued diffusion processes as a noncommutative stochastic process.
• To resolve Biane's open problem on the convergence of Brownian motion on GL_N by identifying the limiting free Itô process.
• To unify free unitary Brownian motion (r=1,s=0) and free multiplicative Brownian motion (r=s=1/2) as special cases of the limiting process.

Proposed method

• Introduces a two-parameter family of left-invariant Riemannian metrics on GL_N via the inner product ⟨ξ₁+iη₁,ξ₂+iη₂⟩_{r,s}^N = -N/r Tr(ξ₁ξ₂) - N/s Tr(η₁η₂) for ξᵢ,ηᵢ ∈ u_N.
• Constructs the associated Laplace-Beltrami operator Δ_{r,s}^N and defines the GL_N Brownian motion B_{r,s}^N(t) as the diffusion with generator (1/2)Δ_{r,s}^N.
• Defines the limiting free Itô process b_{r,s}(t) as the solution to the free SDE db_{r,s}(t) = b_{r,s}(t) dw_{r,s}(t) - (1/2)(r-s)b_{r,s}(t) dt, with w_{r,s}(t) = i√r x(t) + √s y(t).
• Uses asymptotic freeness of matrix increments and moment convergence to prove that the noncommutative distribution of B_{r,s}^N(t) converges to that of b_{r,s}(t) as N → ∞.
• Applies a trace polynomial decomposition and intertwining formula to relate matrix moments to free moments via the P^{j,k}_τ functional.
• Employs a partitioning of time intervals and increment decomposition to reduce joint moment convergence to free independence and stationarity of increments.

Experimental results

Research questions

• RQ1 What is the large-N limit of the Brownian motion on GL_N equipped with a two-parameter family of invariant Riemannian metrics?
• RQ2 How does the limiting process relate to known free stochastic processes such as free unitary and free multiplicative Brownian motion?
• RQ3 Can the convergence of the noncommutative distribution of GL_N Brownian motion be established rigorously using asymptotic freeness and moment methods?
• RQ4 What is the precise form of the free stochastic differential equation that governs the limiting process b_{r,s}(t)?
• RQ5 How do the parameters r and s interpolate between different types of free Brownian motions in the large-N limit?

Key findings

• The noncommutative distribution of the GL_N Brownian motion B_{r,s}^N(t) converges to that of the free Itô process b_{r,s}(t) as N → ∞, for all r,s > 0.
• The limiting process b_{r,s}(t) satisfies the free SDE db_{r,s}(t) = b_{r,s}(t) dw_{r,s}(t) - (1/2)(r-s)b_{r,s}(t) dt with w_{r,s}(t) = i√r x(t) + √s y(t), where x(t), y(t) are freely independent free semicircular Brownian motions.
• When (r,s) = (1,0), the limiting process b_{1,0}(t) is the free unitary Brownian motion.
• When r = s = 1/2, the limiting process b_{1/2,1/2}(t) is the free multiplicative Brownian motion.
• The convergence is established via moment matching: lim_{N→∞} E[tr(g(B_{t₁}^{ε₁}⋯B_{tₙ}^{εₙ}))] = τ(g(b_{t₁}^{ε₁}⋯b_{tₙ}^{εₙ}))) for any noncommutative polynomial g.
• The increments of B_{r,s}^N(t) are asymptotically free and stationary, which enables the reduction of joint moment convergence to free independence of the limiting increments.