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Supersymmetric Analysis of a Random Band Matrix at Criticality, Volume I

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Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We study the N×NN \times N complex Hermitian Gaussian band matrix whose variance profile is the inverse of I−W2ΔI - W^2 \Delta, where Δ\Delta is the Neumann Laplacian on {1,…,N}\{1, \dots, N\}, in the critical regime W2≍NW^2 \asymp N. We prove that, for energies EE in a neighborhood of zero, the unfolded eigenvalue process converges to a stationary point process, whose law depends only on the limit XX of N/((4−E2)W2)N/((4 - E^2) W^2) and is given by the partition functions of a supersymmetric sigma model on an interval. We also determine the macroscopic law of the mass of an eigenvector selected near a given energy. Its density is the normalized exponential of a Brownian motion with drift, under the law tilted by its total mass; the same law describes the eigenvectors of the critical one-dimensional random Schrödinger operator. Further, we compute the limits of the two-energy correlations of the local densities of states at all pairs of interior sites, of the mean eigenvector profile at the scale of single sites, and of NN times the mean inverse participation ratio, which is 2+2X/32 + 2X/3. As X→0X \to 0 and X→∞X \to \infty, we recover the pair statistics of the GUE and of a Poisson process, and in the localized limit we obtain an explicit random shape for the eigenvectors. The number variance in a window of length LL grows like XL\sqrt{XL} as L→∞L \to \infty at every fixed XX, and we describe the crossover to Poisson statistics in windows of length of order XX. Several of these results confirm predictions that physicists obtained from the sigma model. The proof starts from the supersymmetric representation of averages of ratios of characteristic polynomials, which for this profile is a transfer operator on a one-dimensional chain. We show that the transfer chain converges to the heat semigroup of the sigma model, whose noncompact direction produces the Brownian motion in the eigenvector law.

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Version history

  1. v1 · 179 pagesSubmitted by Patrick LopattoInitial depositSupersededNov 12, 2026
  2. v2 · 179 pagesSubmitted by Patrick LopattoSubmitted as a new versionCurrentNov 13, 2026