We investigate the method of moments for $d$-regular digraphs and the limiting $d$-regular directed tree $T_d$ as the number of vertices tends to infinity, in the same spirit as McKay (Linear Algebra Appl., 1981) for the undirected setting. In particular, we provide a combinatorial derivation of the formula for the star moments (from a root vertex $o\in T_d$) $$M_d(w)\qquad:=\sum_{\substack{v_0,v_1\ldots,v_{k-1},v_k\in T_d\\v_0=v_k=o}}A^{w_1}(v_0,v_1)A^{w_2}(v_1,v_2) \cdots A^{w_k}(v_{k-1},v_k)$$ with $A$ the adjacency matrix of $T_d$, where $w:=w_1\cdots w_k$ is any word on the alphabet $\{1,∗\}$ and $A^∗$ is the adjoint matrix of $A$. Our analysis highlights a connection between the non-zero summands of $M_d(w)$ and the non-crossing partitions of $\{1,\ldots,k\}$ which are in some sense compatible with $w$.

It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko−Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas of the logarithmic energy of the mentioned models which can be of independent interest.

We study the first and second orders of the asymptotic expansion, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the $p$-Schatten unit ball. We consider the case of matrices with real, complex or quaternionic entries, self-adjoint or not. When $p>3$, this asymptotic expansion allows us to establish a generalized version of the variance conjecture for the family of $p$-Schatten unit balls of self-adjoint matrices.

We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two noncommutative probability subspaces $M_1$ and $M_2$ and show that the maximal correlation coefficient between the sub-algebras generated by $s_n:=x_1+\ldots+x_n$ and $s_m:=x_1+\ldots +x_m$ equals $\sqrt{m/n}$ for $m\le n$, where $(x_i)_{i\in \mathbb{N}}$ is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo--Kagan--Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotonicity of the free entropy and free Fisher information in the free central limit theorem. Moreover, we prove that the free Stein Discrepancy introduced by Fathi and Nelson is non-increasing along the free central limit theorem.

We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth-fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.

Markovian growth-fragmentation processes introduced by Bertoin extend the pure-fragmentation model by allowing the fragments to grow larger or smaller between dislocation events. What becomes of the known asymptotic behaviors of self-similar pure fragmentations when growth is added to the fragments is a natural question that we investigate in this paper. Our results involve the terminal value of some additive martingales whose uniform integrability is an essential requirement. Dwelling first on the homogeneous case, we exploit the connection with branching random walks and in particular the martingale convergence of Biggins to derive precise asymptotic estimates. The self-similar case is treated in a second part; under the so called Malthusian hypotheses and with the help of several martingale-flavored features recently developed by Bertoin et al., we obtain limit theorems for empirical measures of the fragments.

In this paper we study the number of key exchanges required by Hoare’s FIND algorithm (also called Quickselect) when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed rank. After normalization we give a limit theorem where the limit law is a perpetuity characterized by a recursive distributional equation. To make the limit theorem usable for statistical methods and statistical experiments we provide an explicit rate of convergence in the Kolmogorov--Smirnov metric, a numerical table of the limit law’s distribution function and an algorithm for exact simulation from the limit distribution. We also investigate the limit law’s density. This case study provides a program applicable to other cost measures, alternative models for the rank selected and more balanced choices of the pivot element such as median-of-$2t+1$ versions of Quickselect as well as further variations of the algorithm.