1
vote
0answers
198 views

bivariate polynomial

Hello, Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex. If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where ...
1
vote
0answers
132 views

Distance between probability amplitude functions

Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$: The Wasserstein distance For ...
1
vote
1answer
220 views

Fourier inversion formula for complex-valued random variables?

The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by $$ \phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu $$ (or, so says Wikipedia). How does one recover the ...
5
votes
3answers
420 views

Traceless GUE : Four Centered Fermions

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel \[ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...