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On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions derives the lower bound (theorem 1) $$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n …

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1/4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ …

Note that the random vector $u=(u_1,u_2,\ldots u_n)$, uniformly distributed on the unit sphere, can be replaced by the ratio $u=y/|y|$, with $y=(y_1,y_2,\ldots y_n)\sim N(0,I_n)$ a multivariate normal …

General remark on why one would study simple models: In the statistical mechanics of phase transitions one distinguishes relevant and irrelevant variables. A phase transition is associated with a dive …

Multi-dimensional generalizations (one player against $d$ other players) are explored by P. Lorek in Generalized Gambler's Ruin Problem: explicit formulas via Siegmund duality. For the analogue on a r …

The Fourier transform of the marginal distribution of a single eigenvalue in the GUE is known, $$f_{{\rm GUE}(d)}(k,0,0,\ldots,0)=e^{-\tfrac{1}{2}k^2/d}\sum_{j=0}^{d-1}(-1)^jk^{2j}\frac{(d-1)(d-2)\cdo …

Denote $p=mn$, then the mean $\bar{k}$ you seek equals $\bar{k}=\frac{p!}{(p-j)!}f_j(p)$, with $$f_j(p)=\sum _{k=1}^{\infty } \frac{k \mathcal{S}_k^{(j)} }{p^{k}}=\frac{F_j(p)}{\prod_{n=1}^j(p-n)^2}. …

A proof is on page 80-81 and 90 of Forrester, the probability distribution function of $W=X^\top X$ is $\propto e^{-\tfrac{1}{2}\operatorname{tr}W}(\operatorname{det}W)^{(n-m-1)/2}$, for an $n\times m …

Use the Euler angle parameterisation of the rotation matrix, $$Z=\begin{pmatrix} R(\alpha)&0\\ 0&1\end{pmatrix} \begin{pmatrix} 1&0\\ 0&R(\theta)\end{pmatrix} \begin{pmatrix} R(\alpha')&0\\ 0&1\end{pm …

You want to think of the Haar measure $d\mu(U)$ as a way of measuring uniformity in the group $U(N)$ of unitary $N\times N$ matrices. To form your intuition, consider $N=1$. You then have $U=e^{i\phi} …

The Hilbert-Polya approach to the Riemann hypothesis follows this path, by attempting to relate the zeroes of the Riemann zeta function to a quantum mechanical scattering problem. The probability dist …

Some pointers to the (extensive) literature on generalized Feyman-Kac formulas: Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian Fractional …

K.C. Locey, Random integer partitions with restricted numbers of parts An algorithm is presented to generate uniform random samples of integer partitions for a total $Q$ with $N$ parts from the set o …

You want to derive the Fokker-Planck equation (the drift-diffusion equation for the density) from the Langevin equation (the stochastic differential equation for the position of a particle); this is s …

For elliptic PDE applications to options these would need be independent of time, they need to be perpetual (i.e. never expire), which is not a typical scenario. If your definition of "mathematical fi …