Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rectangular matrix additions in low and high temperatures

Jiaming Xu

Abstract

We study the addition of two random independent $M\times N$ rectangular random matrices with invariant distributions in two limit regimes, where the parameter beta (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers for the empirical measures. As a consequence, we deliver a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and degenerate to the classical or free cumulants in special cases.

Rectangular matrix additions in low and high temperatures

Abstract

We study the addition of two random independent rectangular random matrices with invariant distributions in two limit regimes, where the parameter beta (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers for the empirical measures. As a consequence, we deliver a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and degenerate to the classical or free cumulants in special cases.
Paper Structure (31 sections, 45 theorems, 270 equations, 2 figures)

This paper contains 31 sections, 45 theorems, 270 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Rectangular matrix addition
  4. Low and high temperature behavior
  5. Duality between low and high temperatures
  6. Techniques and difficulties
  7. Further Studies
  8. Bessel functions and Dunkl operators
  9. Symmetric polynomials
  10. Type BC Bessel functions
  11. Type BC Dunkl operators
  12. Matrix addition and moments
  13. Type BC Bessel generating functions
  14. Concentration in low temperature
  15. Finite Law of Large Numbers
  16. ...and 16 more sections

Key Result

Theorem 1.2

B1 Define $\{A_{M}\}_{M=1}^{\infty}$, $\{B_{M}\}_{M=1}^{\infty}$ as above. Assume that $M,N\rightarrow \infty$ in a way that $N(M)/M\rightarrow q$ for some constant $q\ge 1$, and there exists deterministic probability measures $\mu_{A}$, $\mu_{B}$ on $\mathbb{R}$, such that Then the random empirical measure of $C$, $\mu_{C_{M}}=\frac{1}{2M}\sum_{i=1}^{M}(\delta_{c_{M,i}}+\delta_{-c_{M,i}})$, conv

Figures (2)

  • Figure 1: The graph on the left represents a noncrossing partition $\pi$ of $[6]$, where $B_{1}=\{1,4,6\}$, $B_{2}=\{2,3\}$, $B_{3}=\{5\}$, and the graph on the right represents a crossing partition $\pi^{'}$ of $[6]$, where $B_{1}=\{1,3,6\}$, $B_{2}=\{2,4\}$, $B_{3}=\{5\}$.
  • Figure 2: The graphical representation of the non -crossing partition in Example \ref{['ex:partition']}.

Theorems & Definitions (131)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • ...and 121 more