Inverse of a Sum of Random Matrices: A Dynamical Mean-Field Approach
Abstract
We study (O⊤ A O + B)−1, where A, B ∈ ℝN × N are symmetric positive definite, with limiting eigenvalue distributions FA, FB and spectra bounded and bounded away from zero, and O is Haar orthogonal. We analyze the deviation of deterministic-equivalent-type approximation Δdoteq (O⊤ A O + B)−1− S−1 where S = B − z⋆ I. First, let O be independent of (A, B, u), |u| = 1. There is sequence of a standard Gaussian vectors gN independent of (A, B, u) with | S Δ S u − √(τ/N) gN | → 0 a.s. Here z⋆ = −R(−χ) and τ= R'(−χ)(1 + ηR'(−χ)), where R is the R-transform of FA and $(χ, η)$ are the first two inverse moments of the free additive convolution of FA, FB. Let K = ∑i ≤ ⌊ Nq ⌋ λi ui vi⊤, q ∈ [0, C], with unit-norm, not necessarily orthogonal ui, vi and limsupN maxi |λi| < ∞ a.s. If O is independent of (A, B, {ui, vi}), then N−q tr(K Δ) → 0 a.s. The proofs use a dynamical mean-field approach: the generating functions of the free Appell polynomials give a linear dynamics approximating the quantity of interest in the iterated large-N, large-time limit, whose fluctuations we analyze.
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BibTeX
@misc{cakmak2026inverse,
title = {{Inverse of a Sum of Random Matrices: A Dynamical Mean-Field Approach}},
author = {Burak Çakmak and Manfred Opper},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21140},
eprint = {2609.21140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21140},
note = {CC0},
}
RIS
TY - UNPB
TI - Inverse of a Sum of Random Matrices: A Dynamical Mean-Field Approach
AU - Burak Çakmak
AU - Manfred Opper
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21140
UR - https://arxiv.org/abs/2609.21140
AB - We study (O^⊤ A O + B)^−1, where A, B ∈ ℝ^N × N are symmetric positive definite, with limiting eigenvalue distributions F_A, F_B and spectra bounded and bounded away from zero, and O is Haar orthogonal. We analyze the deviation of deterministic-equivalent-type approximation Δdoteq (O^⊤ A O + B)^−1− S^−1 where S = B − z^⋆ I. First, let O be independent of (A, B, u), |u| = 1. There is sequence of a standard Gaussian vectors g_N independent of (A, B, u) with | S Δ S u − √(τ/N) g_N | → 0 a.s. Here z^⋆ = −R(−χ) and τ= R'(−χ)(1 + ηR'(−χ)), where R is the R-transform of F_A and $(χ, η)$ are the first two inverse moments of the free additive convolution of F_A, F_B. Let K = ∑_i ≤ ⌊ N^q ⌋ λ_i u_i v_i^⊤, q ∈ [0, C], with unit-norm, not necessarily orthogonal u_i, v_i and limsup_N max_i |λ_i| < ∞ a.s. If O is independent of (A, B, {u_i, v_i}), then N^−q tr(K Δ) → 0 a.s. The proofs use a dynamical mean-field approach: the generating functions of the free Appell polynomials give a linear dynamics approximating the quantity of interest in the iterated large-N, large-time limit, whose fluctuations we analyze.
ER -