Related papers: $S$-transform in Finite Free Probability
We study the properties of the set of marginal distributions of infinite translation-invariant systems in the 2D square lattice. In cases where the local variables can only take a small number $d$ of possible values, we completely solve the…
Denote by $J$ the operator of coefficient stripping. We show that for any free convolution semigroup of measures $\nu_t$ with finite variance, applying a single stripping produces semicircular evolution with non-zero initial condition,…
Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to…
The paper studies upper bounds for the total variation distance between two polynomials of a special form in random vectors satisfying the Doeblin-type condition. Our approach is based on the recent results concerning Nikolskii--Besov-type…
We develop further the graph limit theory for dense weighted graph sequences. In particular, we consider probability graphons, which have recently appeared in graph limit theory as continuum representations of weighted graphs, and we…
We give a Banach algebra approach to the additiveness property of Voiculescu's amalgamated R-transform. We also define an amalgamated S-transform and prove that it is multiplicative on products of amalgamated free random variables when the…
We give a streamlined proof of the limit theorems for the free additive convolution of infinitesimal triangular arrays of probability measures on the real line. The result was first proved by Chistyakov and G\"otze using analytic…
We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other…
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group…
We study the behavior of the tail of a measure $\mu^{\boxtimes t}$, where $\boxtimes t$ is the $t$-fold free multiplicative convolution power for $t\geq 1$. We focus on the case where $\mu$ is a probability measure on the positive half-line…
Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the…
According to the Dudley-Wichura extension of the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s. in the metric. A density…
Hoffstein and Hulse defined the shifted convolution series of two cusp forms by "shifting" the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of…
Recent Monte Carlo simulations of a grafted semiflexible polymer in 1+1 dimensions have revealed a pronounced bimodal structure in the probability distribution of the transverse (bending) fluctuations of the free end, when the total contour…
The paper is a sketch of systematic presentation of distributional limit theorems and their refinements for compound sums. When analyzing, e.g., ergodic semi-Markov systems with discrete or continuous time, this allows us to separate those…
We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $p=1$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique…
In a recent paper the author obtained optimal bounds for the strong Gaussian approximation of sums of independent $\R^d$-valued random vectors with finite exponential moments. The results may be considered as generalizations of well-known…
We show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions…
We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution…
We derive sharp upper and lower bounds for the pointwise concentration function of the maximum statistic of $d$ identically distributed real-valued random variables. Our first main result places no restrictions either on the common marginal…