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This research aims to demonstrate the convergence of eigenvector processes associated with generalized Wigner matrices.

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June 6, 2026

Convergence of local eigenvector processes of generalized Wigner matrices

Key Points

This research aims to demonstrate the convergence of eigenvector processes associated with generalized Wigner matrices.
Proved convergence using multi-dimensional convergence techniques from [BC24].
Applied a tightness criterion based on Hölder regularity of observables.
Provided explicit examples of limiting processes and their relationships with Gaussian processes.
Identified a large class of Gaussian processes with Hölder-continuous covariance as limits.
Established that eigenvector processes converge to these limiting Gaussian processes.
Showed that limits can be represented using Karhunen–Loéve expansion techniques.

Abstract

We prove convergence of eigenvector processes of the form Formula: see text where uk is a bulk eigenvector of generalized Wigner matrices and (Formula: see text) is a family of symmetric matrices with bounded norm and Hölder regularity. We give explicit examples of limiting processes and prove that a large class of Gaussian processes with Hölder-continuous covariance function can be obtained as such a limit using its Karhunen–Loéve expansion. The proof is based on the multi-dimensional convergence proved in BC24 and a tightness criterion proved using Hölder regularity of the observables.

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Convergence of local eigenvector processes of generalized Wigner matrices

Key Points

Abstract

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