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The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the podiu …

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well define …

Consider $N$ real random particles $x_1,\cdots, x_N$ distributed according to a density $\rho(x_1,\ldots,x_N)$ with respect to the Lebesgue measure on $\mathbb R^N$, which is assumed to be invariant u …

You can rewrite $$ S=\frac{1}{M}C^{1/2}XX^T(C^{1/2})^T $$ where $X_{.j}\sim \mathcal{N}(\boldsymbol 0, \boldsymbol I)$. It is then classical that the limiting eigenvalue distribution of $S$ (in the …

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with law $\mu$. The Sanov Theorem then states that the empirical measures $$ \mu^N =\frac{1}{N} \sum _{n=1}^N\delta _{X_n} $$ sat …

These three ensembles are hermitian matrices over a (finite dimensional real) field of numbers, and it is known that the only finite dimensional real fields are the real numbers, the complex numbers ( …

We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi_*:M_1(\mathbb R^2)\rightarrow M_1(\mathbb R)$ the induced push-forward map (where $M_1(\Delta)$ stands fo …

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, an …

By simple computations : The definition of the modified Bessel function of the first kind yields $$ I_k(\lambda)=\sum_{n\geq 0}\frac{1}{n!(n+k)!}\left(\frac{\lambda}{2}\right)^{2n+k} $$ so that we ge …

I would like to have a better understanding of a notion I've met in the beautiful book of Nikishin and Sorokin "Rational approximation and Orthogonality", since they do not provide examples. As it is …

It seems that what you are looking for is the notion of "abstract Wiener space", which provides a rigorous setting for putting a white noise on various spaces of functions. Long story short, you defin …

Yes, you can define properly the first expectation, see e.g. PlanetMath Then you have with your notations $$ \mathbb E_{X_i : i\in\mathbb N}\int_{[0,1]^k}\inf_{i\in\mathbb N}\|X_i-y\|^2dy\leq \math …

If you accept non-commutative random variables as "other random processes $(X_1,\ldots,X_n)$" then, as Jon Bannon's comment suggests, free probability could provide interesting examples [and I don't t …

Just an heuristic answer: The joint eigenvalue distribution of a GOE random matrix is given by $$ \frac{1}{Z_n}\prod_{i<j}|x_i-x_j|\prod_{i=1}^ne^{-x_i^2/2}dx_i. $$ Thus, $E_{GOE}(|\det(y+A)|e^{-x(tr …

I understand that you consider a "parametrized" identity on a space of differentiable functions $$ \mathcal{L}_{a,b}(f,g)=\[f-a\]\[g-b\]+a\[g-b\]+b\[f-a\]-fg+ab=0 $$ and two different linear forms act …