Related papers: $S$-transform in Finite Free Probability
An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A…
In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $P = (P_1,\dots, P_n)$ of functions $P_i \colon…
We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…
We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,\infty), called the free regular measures. We…
We prove that $X^r$ follows an FID distribution if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r\in(-\infty,0]\cup[1,\infty)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r\geq1$; (3) $X$…
We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic L\"owner Evolution methods. These quantities are shown to…
The present material addresses several problems left open in the Trans. AMS paper " Non-crossing cumulants of type B" of P. Biane, F. Goodman and A. Nica. The main result is that a type B non-commutative probability space can be studied in…
Bercovici and Pata showed that the correspondence between classically, freely, and Boolean infinitely divisible distributions holds on the level of limit theorems. We extend this correspondence also to distributions infinitely divisible…
We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the…
Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the…
The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the…
Multiplicative self-decomposable laws describe random variables that can be decomposed into a product of a scaled-down version of themselves and an independent residual term. Shanbhag et al.~(1977) have shown that the gamma distribution is…
We introduce a theory of probability in $\lambda$-rings designed to efficiently describe random variables valued in multisets of complex numbers, varieties over a field, or other similar enriched settings. A key role is played by the…
We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane…
The study of convolution powers of a finitely supported probability distribution $\phi$ on the $d$-dimensional square lattice is central to random walk theory. For instance, the $n$th convolution power $\phi^{(n)}$ is the distribution of…
In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D.…
We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order…
Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by…
The asymptotic study of tuples of random non-increasing integers is crucial for probabilistic models coming from asymptotic representation theory and statistical physics. We study the global behavior of such tuples, introducing a new family…
We introduce two 2-variables transforms: the partial bi-free S-transform and the partial bi-free T-transform. These transforms are the analogues for the bi-multiplicative and respectively for the additive-multiplicative bi-free convolution…