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Operator Norm Bounds for Multi-leg Matrix Tensors and Applications to Random Matrix Theory

Benoît Collins, Wangjun Yuan

Abstract

We investigate the extremal values of partial traces of matrix tensors under operator norm constraints. To evaluate these multi-linear quantities, we develop a comprehensive graphical formalism that encodes multi-leg partial traces, partial permutations, and their moments using colored directed graphs. With this graphical framework, we establish optimal, sharp bounds for the partial trace $(\mathrm{Tr}_{σ_1} \otimes \ldots \otimes \mathrm{Tr}_{σ_k})(A_1, \ldots, A_m)$ over matrices bounded by $\|A_i\| \le 1$. Specifically, we prove that this maximum evaluates exactly to $N^{M(σ_1,\ldots,σ_k)}$, where $N$ is the dimension and $M$ represents the maximal number of directed cycles in the associated graph across all possible internal vertex pairings. We further derive explicit operator norm estimates for matrices generated by partial traces of partial permutations. Finally, we apply these combinatorial bounds to multi-matrix random matrix theory. By examining models involving Ginibre ensembles, we extend concepts of asymptotic freeness to matrix coefficient algebras, establishing operator norm estimates that rigorously separate the asymptotic behavior of non-crossing and crossing pairings.

Operator Norm Bounds for Multi-leg Matrix Tensors and Applications to Random Matrix Theory

Abstract

We investigate the extremal values of partial traces of matrix tensors under operator norm constraints. To evaluate these multi-linear quantities, we develop a comprehensive graphical formalism that encodes multi-leg partial traces, partial permutations, and their moments using colored directed graphs. With this graphical framework, we establish optimal, sharp bounds for the partial trace over matrices bounded by . Specifically, we prove that this maximum evaluates exactly to , where is the dimension and represents the maximal number of directed cycles in the associated graph across all possible internal vertex pairings. We further derive explicit operator norm estimates for matrices generated by partial traces of partial permutations. Finally, we apply these combinatorial bounds to multi-matrix random matrix theory. By examining models involving Ginibre ensembles, we extend concepts of asymptotic freeness to matrix coefficient algebras, establishing operator norm estimates that rigorously separate the asymptotic behavior of non-crossing and crossing pairings.

Paper Structure

This paper contains 23 sections, 29 theorems, 92 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Graphical notation
  3. Graph for 2 legs
  4. Graph for multiple legs
  5. Graphs for partial permutations
  6. Graphs for moments
  7. Main results
  8. Partial trace with 2 legs
  9. Partial trace with multiple legs
  10. Matrix of partial trace for partial permutations
  11. Upper bound in the 2 legs case
  12. Upper bound for partial trace
  13. Proof of Theorem \ref{['Thm-upper']}
  14. Lower bound in the 2 legs case
  15. Lower bound for partial trace
  16. ...and 8 more sections

Key Result

Theorem 1

For any integer $k \ge 1$ and permutations $\sigma_1, \ldots, \sigma_k \in \mathcal{P}([m])$, the maximum of the multi-leg partial trace over matrices bounded in operator norm is governed exactly by the cycle structure of the graph:

Figures (21)

  • Figure 1: Graph $G_{\sigma_1,\sigma_2} (A_1,\ldots,A_m)$ with $m=4$ and $\sigma_1=(1,2,3)(4)$, $\sigma_2=(1,2,3,4)$
  • Figure 2: Blue directed edges in graph $G_{\sigma_1,\sigma_2} (A_1,\ldots,A_m)$ with $m=4$ and $\sigma_1=(1,2,3)(4)$, $\sigma_2=(1,2,3,4)$
  • Figure 3: Graph $G_{\sigma_1,\ldots,\sigma_4} (A_1,\ldots,A_4)$ with $\sigma_1=\sigma_2=(123)(4)$ and $\sigma_3=\sigma_4=(1234)$
  • Figure 4: $k=m=3$, graph $G_{\sigma_1,\sigma_2,\sigma_3} (A_1,\ldots,A_3)$, $\sigma_2=\sigma_3=(123)$ and $\sigma_1(1)=2, \sigma_1(2)=3$
  • Figure 5: $k=m=3$, graph $G^*_{\sigma_1,\sigma_2,\sigma_3}(A_1^*,A_2^*,A_3^*)$ with $\sigma_2=\sigma_3=(123)$ and $\sigma_1(1)=2, \sigma_1(2)=3$
  • ...and 16 more figures

Theorems & Definitions (57)

  • Theorem : cf. Theorem \ref{['Thm-main-multi']}
  • Theorem : cf. Theorem \ref{['Thm-main-matrix']} and Corollary \ref{['Coro-operator norm']}
  • Theorem : cf. Theorems \ref{['Thm-Ginibre-1']} and \ref{['Thm-Ginibre-2']}
  • Remark 1
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Remark 3
  • Theorem 2
  • ...and 47 more