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Incremental universality of Wigner random matrices

Giovanni M. Cicuta, Mario Pernici

TL;DR

This work formalizes incremental universality for Wigner random matrices, showing that the large-$n$ behavior of connected correlators is governed by matched matrix-entry moments in a stepwise fashion. By leveraging a walk-based combinatorial framework and Green’s-function formalism, it derives explicit $1/n$ expansions for single-, double-, and higher-order trace correlators, with higher moments $v_{2j}$ entering only at specific orders determined by how many moments are matched. The results reveal that universality is hierarchical: ensembles sharing moments up to $v_{2j-2}$ exhibit identical leading corrections up to order $n^{-(j-1)}$, while differences at $v_{2j}$ appear with calculable coefficients and spectral-density counterparts via $R_j(y)$ functions. The paper provides exact low-order correlator formulas and a comprehensive appendix suite that encodes the combinatorial machinery, offering a detailed blueprint for assessing universality across diverse Wigner-like ensembles. This advances the understanding of how universal spectral statistics emerge from moment matching in non-invariant random-matrix models and clarifies the role of higher moments in shaping fluctuations.

Abstract

Properties of universality have essential relevance for the theory of random matrices usually called the Wigner ensemble. The issue was analysed up to recent years with detailed and relevant results. We present a slightly different view and compare the large-$n$ limit of connected correlators of distinct ensembles: universality has steps or degrees, precisely counted by the number of probability moments of the matrix entries, which match among distinct ensembles.

Incremental universality of Wigner random matrices

TL;DR

This work formalizes incremental universality for Wigner random matrices, showing that the large- behavior of connected correlators is governed by matched matrix-entry moments in a stepwise fashion. By leveraging a walk-based combinatorial framework and Green’s-function formalism, it derives explicit expansions for single-, double-, and higher-order trace correlators, with higher moments entering only at specific orders determined by how many moments are matched. The results reveal that universality is hierarchical: ensembles sharing moments up to exhibit identical leading corrections up to order , while differences at appear with calculable coefficients and spectral-density counterparts via functions. The paper provides exact low-order correlator formulas and a comprehensive appendix suite that encodes the combinatorial machinery, offering a detailed blueprint for assessing universality across diverse Wigner-like ensembles. This advances the understanding of how universal spectral statistics emerge from moment matching in non-invariant random-matrix models and clarifies the role of higher moments in shaping fluctuations.

Abstract

Properties of universality have essential relevance for the theory of random matrices usually called the Wigner ensemble. The issue was analysed up to recent years with detailed and relevant results. We present a slightly different view and compare the large- limit of connected correlators of distinct ensembles: universality has steps or degrees, precisely counted by the number of probability moments of the matrix entries, which match among distinct ensembles.
Paper Structure (17 sections, 81 equations)

This paper contains 17 sections, 81 equations.

Table of Contents

  1. Introduction
  2. Incremental universality of the single-trace averages.
  3. $r$-trace connected correlators for $r \ge 2$
  4. Conclusions
  5. Supporting information Appendix, Introduction and Index
  6. Appendix A: Connected correlators and walks
  7. Review of connected correlators
  8. Enumeration of walks
  9. Appendix B: Leading order of the $r$-trace connected correlator contributions containing $v_{2j}$
  10. Properties of $T$
  11. Path enumeration
  12. Appendix C: $1/n$-order one-point Green function
  13. Appendix D: Leading order of the two-point connected correlation function
  14. Appendix E: Exact connected correlators at low orders.
  15. Single-trace averages.
  16. ...and 2 more sections