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I could go back to Markov himself, who in 1913 applied the concept of a Markov chain to sequences of vowels and consonants in Alexander Pushkin's poem Eugene Onegin. In good approximation, the probabi …

• Mathoverflow has been studied as a "complex network" in Social achievement and centrality in MathOverflow, by L.V. Montoya, A. Ma, and R.J. Mondragón. The analysis distinguishes degree centrality (b …

It might help to take a broader perspective: in some contexts (notably quantum optics) the emphasis is not on cumulants but on factorial cumulants, with generating function $h(t)=\log E(t^X)$. While c …

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 comments list many reasons why the Gaussian distribution is special, but is it "the most fundamental" among all distributions, as suggested in the OP? I would like to argue that (1) conservation l …

If you choose the matrix elements of $A$ independently from a Gaussian distribution you have the socalled Ginibre ensemble of random-matrix theory. The statistics of the eigenvalues is known, see for …

it is positive for $m=1$, but not for $m=2$, see this Mathematica output:

This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized Sums (1973). The Cauchy distribution is the case $\alpha=1$ on page 798. The singulari …

The physics application I am aware of is not quite the one in the OP, but similar in spirit: in ray optics the SL(2,R) matrix $$g=\begin{pmatrix} A & B \\ C & D \end{pmatrix}$$ describes the effect …

Q: Am I overlooking something important? I think you are ignoring the role played by the thermodynamic limit. There are two interplaying limits here, the replica limit $n\rightarrow 0$ and the thermod …

The earliest reference I have found for this result is Entropy and maximal spacings for random partitions (E. Slud, 1978). Theorem 2.2 states that the entropy $W_n=-\sum_{i=1}^n p_i \ln p_i$ of the r …

well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, you could treat this problem as the special case $n=2$ of a classic problem in random-matrix theory: take an $n\times n …

The only rigorous bound I am aware of is due to Gonzalo Navarro* $$c\geq 1-{\rm e}/\sqrt{\sigma},$$ for an alphabet of $\sigma$ characters. Obviously, for the binary string ($\sigma=2$) this bound i …

This is the problem of Brownian motion between two moving absorbing boundaries. For a linear time dependence some analytical progress can be made, but for arbitrary time dependence no closed-form solu …

This may be a less "clear-cut-no" than suggested by the mantra "there can be no massive particles in a CFT because that would introduce a scale": Anatol Odzijewicz has constructed a CFT for massive pa …