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Limit joint distributions of SYK Models with partial interactions, Mixed q-Gaussian Models and Asymptotic $\varepsilon$-freeness

Weihua Liu, Haoqi Shen

TL;DR

This work analyzes the joint distribution of multiple SYK Hamiltonians with prescribed overlaps and proves convergence, in the large-$n$ limit, to a mixed $Q$-Gaussian system with $q_{ij}=(-1)^{r_i r_j} e^{-2\lambda_{ij}}$, where $\lambda_{ij}$ encodes pairwise overlaps. The authors realize ε-free independence through graph-product constructions, yielding a random model for asymptotic $\varepsilon$-freeness. By connecting SYK models to deformed Fock spaces and operator-algebraic independence, the paper provides a concrete probabilistic framework for noncommutative independence in quantum spin systems and broadens the applicability of mixed $q$-Gaussian variables. The results illuminate how interaction length and subsystem overlaps govern noncommutativity and enable explicit limiting moments via pairings, contributing a rigorous bridge between random matrix models and $\varepsilon$-freeness in free probability.

Abstract

We study the joint distribution of SYK Hamiltonians for different systems with specified overlaps. We show that, in the large-system limit, their joint distribution converges in distribution to a mixed $q$-Gaussian system. We explain that the graph product of diffusive abelian von Neumann algebras is isomorphic to a $W^*$-probability space generated by the corresponding $\varepsilon$-freely independent random variables with semicircular laws which form a special case of mixed $q$-Gaussian systems that can be approximated by our SYK Hamiltonian models. Thus, we obtain a random model for asymptotic $\varepsilon$-freeness.

Limit joint distributions of SYK Models with partial interactions, Mixed q-Gaussian Models and Asymptotic $\varepsilon$-freeness

TL;DR

This work analyzes the joint distribution of multiple SYK Hamiltonians with prescribed overlaps and proves convergence, in the large- limit, to a mixed -Gaussian system with , where encodes pairwise overlaps. The authors realize ε-free independence through graph-product constructions, yielding a random model for asymptotic -freeness. By connecting SYK models to deformed Fock spaces and operator-algebraic independence, the paper provides a concrete probabilistic framework for noncommutative independence in quantum spin systems and broadens the applicability of mixed -Gaussian variables. The results illuminate how interaction length and subsystem overlaps govern noncommutativity and enable explicit limiting moments via pairings, contributing a rigorous bridge between random matrix models and -freeness in free probability.

Abstract

We study the joint distribution of SYK Hamiltonians for different systems with specified overlaps. We show that, in the large-system limit, their joint distribution converges in distribution to a mixed -Gaussian system. We explain that the graph product of diffusive abelian von Neumann algebras is isomorphic to a -probability space generated by the corresponding -freely independent random variables with semicircular laws which form a special case of mixed -Gaussian systems that can be approximated by our SYK Hamiltonian models. Thus, we obtain a random model for asymptotic -freeness.
Paper Structure (11 sections, 15 theorems, 143 equations)

This paper contains 11 sections, 15 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Mixed $q$-Gaussian System
  3. $\epsilon-$ free independence
  4. Mixed q-Gaussian system and $\varepsilon$-free independence
  5. SYK Models
  6. Proofs of technical lemmas
  7. Proof of \ref{['lem:singular']}
  8. Estimate for $a_{p}$
  9. Estimate for $a_{q}$
  10. Proof of \ref{['lem: n-poisson']}
  11. Proof of Proposition \ref{['Main Proposition']}

Key Result

Theorem 1.1

Let $\mathcal{I}$ be an index set. For each $k\in \mathcal{I}$ and $n\in \mathbb{N}$, let $H_{k,n}$ be the SYK model such that where variable $J_{k;i_1,\dots,i_{r_{k,n}}}$ are independent random variables with $\mathbb{E}[J_{k;i_1,\dots,i_{r_{k,n}}}]=0$ and $\mathbb{E}[J_{k;i_1,\dots,i_{r_{k,n}}}^2]=1$, and that for each integer $p\ge 3$ there exists a constant $C_p<\infty$ such that and $\psi_{

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2
  • ...and 16 more