Matrix Addition and the Dunkl Transform at High Temperature

Abstract

We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to infinity simultaneously with the beta (inverse temperature) parameter going to zero. Our approach is based on the analysis of the (symmetric) Dunkl transform in this regime. As an application we obtain the LLN for the sums of random matrices as the inverse temperature goes to 0. This results in a one-parameter family of binary operations which interpolates between classical and free convolutions of the probability measures. We also introduce and study a family of deformed cumulants, which linearize this operation.

9. Appendix: Law of Large Numbers for Fixed Temperature

The aim of this “Appendix” is to probe the possibility of a version of Theorem 3.8 in which $\theta > 0$ is fixed and is not changing with N. The following claim is an analogue of one direction of Theorem 3.8.

Claim 9.1

(LLN for finite temperature). Let $\{\mu _N\}_{N \ge 1}$ be a sequence of exponentially decaying probability measures on tuples $a_1\le \cdots \le a_N$. For each N, let $G_{N; \theta }(x_1, \ldots , x_N) := G_\theta (x_1, \ldots , x_N; \mu _N)$ be the BGF of $\mu _N$. Assume that the sequence $\{G_{N; \theta }\}_N$ satisfies the following conditions:

• (a) $\displaystyle \lim _{N\rightarrow \infty }\left. \frac{1}{N}\cdot \frac{\partial ^l}{\partial x_i^l}\ln {(G_{N; \theta })}\right| _{x_1=\dots =x_N=0} = (l - 1)!\cdot c_l$, for all $l\in \mathbb {Z}_{\ge 1}$.

• (b) $\displaystyle \lim _{N\rightarrow \infty }\left. \frac{1}{N}\cdot \frac{\partial }{\partial x_{i_1}}\cdots \frac{\partial }{\partial x_{i_r}}\ln {(G_{N; \theta })}\right| _{x_1=\dots =x_N=0} = 0$, for all $i_1, \dots , i_r\in \mathbb {Z}_{\ge 1}$ such that the set $\{i_1, \dots , i_r\}$ contains at least two distinct indices.

Then the sequence $\{\mu _N\}_{N \ge 1}$ satisfies the following LLN (compare to Definition 3.1): there exist real numbers $\{m_{k}\}_{k\ge 1}$ such that for any $s=1,2,\dots$ and any $k_1, \dots , k_s\in \mathbb {Z}_{\ge 1}$, we have

$$\begin{aligned} \lim _{N\rightarrow \infty } \mathbb {E}_{\mu _N} \prod _{i=1}^s \left( \frac{1}{N} \sum _{i=1}^N\left( \frac{a_i}{N} \right) ^{k_i} \right) = \prod _{i=1}^{s} m_{k_i}.\end{aligned}$$

If this occurs, $\{c_l\}_{l\ge 1}$ and $\{m_k\}_{k\ge 1}$ are related by either

$$\begin{aligned} m_k = \sum _{\pi \in NC(k)}\, \prod _{B\in \pi }{\left( \theta ^{|B| - 1}c_{|B|} \right) },\quad k = 1, 2, \ldots , \end{aligned}$$

or, equivalently,

$$\begin{aligned} m_k = \frac{1}{k+1} \cdot [z^{-1}] \left( \left( z^{-1} + \sum _{l = 1}^{\infty } \theta ^{l-1}c_l z^{l - 1} \right) ^{\!\!k+1} \right) \!,\quad k\in \mathbb {Z}_{\ge 1}. \end{aligned}$$

We do not present a proof of the claim, but probably one can prove it with the same techniques that we have used in Sect. 5. At $\theta =1$, another approach is by a degeneration of [BuG15, Theorem 5.1], see also [MN18]. This claim would prove the one-sided implication

$$\begin{aligned} \text {conditions (a) and (b) (fixed}\, \theta \,\text { version of LLN-appropriateness)} \Longrightarrow \text {LLN-satisfaction}. \end{aligned}$$

Based on our Theorem 3.8, on [BuG19, Theorem 2.6] (which studies the CLT at fixed $\theta = 1$), and on the classical theorem that relates the weak convergence of measures to the convergence of their characteristic functions, the reader may be inclined to believe that the reverse implication is also true and that this kind of “if and only if” results are always expected.

However, this turns out to be wrong. The naive analogue of Theorem 3.8 does not hold for fixed $\theta$: the “expected if and only if statement" is false. Here is a counter-example.

Let us consider a sequence of probability measures $\mu _N$ such that a random $\mu _N$-distributed vector $a_1 \ge \cdots \ge a_N$ has $a_1 = \cdots = a_N$ almost surely and this common value a is distributed according to a Gaussian measure of mean 0 and variance N. In this case, the random variable

$$\begin{aligned} p_k^N = \frac{1}{N}\sum _{i=1}^N{\left( \frac{a_i}{N} \right) ^k} = \left( \frac{a}{N} \right) ^k \end{aligned}$$

is distributed as the k-th power of a Gaussian random variable of mean 0 and variance 1/N. Consequently, the sequence $\{\mu _N\}_N$ satisfies a LLN, and all $m_k$’s are equal to zero. On the other hand, by using $B_{(a, \dots , a)}(x_1, \dots , x_N; \theta ) = \exp (a\sum _{i=1}^N{x_i})$, it follows that the BGF of $\mu _N$ equals

$$\begin{aligned} G_{N; \theta }(x_1, \ldots , x_N) {=} \int _{{-}\infty }^{\infty }{\frac{e^{{-}a^2/(2N)}}{\sqrt{2\pi N}} B_{(a, \dots , a)}(x_1, \dots , x_N; \theta ) da} {=} \exp \left( \frac{N}{2} \left( \sum _{i=1}^N{x_i}\right) ^{\!\!\!2\,} \right) . \end{aligned}$$

The sequence $\{G_{N; \theta }\}_N$ of BGFs then satisfies

$$\begin{aligned} \lim _{N\rightarrow \infty } \left. \frac{1}{N}\cdot \frac{\partial ^2}{\partial x_1\partial x_2}\ln (G_{N;\theta }) \right| _{x_1=\dots =x_N=0} = 1, \end{aligned}$$

therefore contradicting condition (b) from Claim 9.1.

A more refined question is whether some “if and only if for LLN” statement holds, if one modifies somehow the conditions (a) and (b) of Claim 9.1. Based on small calculations (obtained when trying to reverse-engineer the proof of Theorem 3.8), it is plausible that the answer is yes.

Indeed, based on Proposition 2.11, we must study the limits of the expressions

$$\begin{aligned} N^{-|\lambda |-\ell (\lambda )}\cdot \left[ \prod _{i=1}^{\ell (\lambda )}{\mathcal {P}_{\lambda _i}}\right] \! G_{N; \theta }, \end{aligned}$$

(9.1)

where $\lambda$ ranges over the set of all partitions of a given size k. We have performed calculations for $k=2, 3$; they indicate that the conditions on second-order derivatives in (a) and (b) from Claim 9.1 should be replaced by:

$$\begin{aligned} \bullet&\lim _{N\rightarrow \infty } \left. \frac{1}{N}\left\{ \frac{\partial ^2}{\partial x_1^2} - \frac{\partial ^2}{\partial x_1\partial x_2} \right\} \ln (G_{N;\theta }) \right| _{x_1=\dots =x_N=0} = \theta ^{-1}\cdot c_2,\\ \bullet&\lim _{N\rightarrow \infty } \left. \frac{1}{N^2} \frac{\partial ^2 \ln (G_{N;\theta })}{\partial x_1\partial x_2} \right| _{x_1=\dots =x_N=0} = 0, \end{aligned}$$

and the conditions on third-order derivatives should be replaced by:

$$\begin{aligned} \bullet&\lim _{N\rightarrow \infty } \left. \frac{1}{N}\left\{ \frac{1}{2}\cdot \frac{\partial ^3}{\partial x_1^3} - \frac{3}{2}\cdot \frac{\partial ^3}{\partial x_1^2\partial x_2} + \frac{\partial ^3}{\partial x_1\partial x_2 \partial x_3} \right\} \ln (G_{N;\theta }) \right| _{x_1=\dots =x_N=0} = \theta ^{-2}\cdot c_3,\\ \bullet&\lim _{N\rightarrow \infty } \left. \frac{1}{N^2}\left\{ \frac{\partial ^3}{\partial x_1^2\partial x_2} - \frac{\partial ^3}{\partial x_1 \partial x_2 \partial x_3} \right\} \ln (G_{N;\theta }) \right| _{x_1=\dots =x_N=0} = 0,\\ \bullet&\lim _{N\rightarrow \infty } \left. \frac{1}{N^3}\frac{\partial ^3 \ln (G_{N;\theta })}{\partial x_1\partial x_2\partial x_3} \right| _{x_1=\dots =x_N=0} = 0. \end{aligned}$$

These relations are much more involved than conditions (a) and (b) from Claim 9.1, or than the conditions from Definition 3.3. What should be the correct “if and only if” relations for $k>3$? This is an interesting open question for future research.