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Let $\mathbf{A}$ be a matrix of size $N\times N$ whose elements $A_{ij}$ (with $1\leq i,j\leq N$) are I.I.D following some distribution. If we set set $\langle A_{ij}\rangle=0$ and $\langle {A_{ij}}^2 …

Background: In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices $\mathbf{J}$ by studying the following integral: \begin{equation} I=\left[ …

Considering this system of integral equations, where $\gamma \in \mathbb{R} $ and $\alpha\in \mathbb{C}$ are the unknown to solve : $$ 1=\int_{-\infty}^{\infty} p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\t …

I am looking to solve the following equation: $$\left(\begin{array}{cc}{ g_{11}} & { g_{12}} \\ { g_{21}} & { g_{22}}\end{array}\right)=\int_{a}^{b} \frac{1}{b-a}\left(\begin{array}{cc}{- g_{22}} & { …

I am integrating the following Gaussian over all possible matrix elements $J_{ij}$: $$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm …

Let $M$ be a real symmetric matrix of size $N$ with its components $M_{ij}$ following a normal distribution centered around 0. Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\ …

I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting dis …

One can define the density of eigenvalues of a $N\times N$ Hermitian random matrix $H$ as: \begin{equation} \rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle \end …

The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral: $$I_\tau=\int \prod_{i, j=1}^{N} d J_{i …

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3? Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its c …

Let $T$ be a fully symmetric tensor of rank $3$ and size $N$. Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that: \begin{equation} \sum_{jk}^NT …