Related papers: Bernoulli variables, classical exclusion processes…
We analyze complexity of financial (and general economic) processes by comparing classical and quantum-like models for randomness. Our analysis implies that it might be that a quantum-like probabilistic description is more natural for…
Asymmetric exclusion processes with locally reversible kinetic constraints are introduced to investigate the effect of non-conservative driving forces in athermal systems. At high density they generally exhibit rheological-like behavior,…
We introduce a general statistical learning theory for processes that take as input a classical random variable and output a quantum state. Our setting is motivated by the practical situation in which one desires to learn a quantum process…
A combinatorial approach to free probability theory has been developped by Roland Speicher, based on the notion of noncrossing cumulants, a free analogue of the classical theory of cumulants in probability theory. We review this theory, and…
In the classical probability in continuous random variables there is no distinguishing between the probability involving strict inequality and non strict inequality. Moreover a probability involves equality collapse to zero without…
The aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new and simple singularity theorems, whose features are different from the classical singularity…
We numerically study the large deviation function of the total current, which is the sum of local currents over all bonds, for the symmetric and asymmetric simple exclusion processes with open boundary conditions. We estimate the generating…
We study elementary classical and quantum dynamics in an information geometric space corresponding to a Bernoulli random variable, extending work by Goehle and Griffin [Chaos, Solitons & Fractals, 188, 115535, (2024)], who study the…
A totally asymmetric exclusion process consisting of classical particles with next-nearest-neighbor interactions has been considered on a 1D discrete lattice with a ring geometry. Using large deviation techniques, we have investigated…
A simple classical, deterministic, local situation violating the Bell inequality is described. The detectors used in the experiment are ideal and the observers who decide which pair of measuring devices to choose for a given pair of…
Free cumulants were introduced as the proper analog of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of…
The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial…
We study current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, we prove a large deviation principle for the empirical current in the symmetric simple exclusion process with rate functional I. We then…
In this paper, a connection between bi-free probability and the theory of non-commutative stochastic processes is examined. Specifically it is demonstrated that the transition operators for non-commutative stochastic processes can be…
We investigate analytical properties of free stable distributions and discover many connections with their classical counterparts. Our main result is an explicit formula for the Mellin transform, which leads to explicit series…
We examine the asymmetric simple exclusion process with open boundaries, a paradigm of driven diffusive systems, having a nonequilibrium steady state transition. We provide a full derivation and expanded discussion and digression on results…
In this work, Bernoulli's Law of Large Numbers, also known as the Golden theorem, has been extended to study the relations between empirical probability and empirical randomness of an otherwise random experiment. Using the example of a coin…
For the asymmetric simple exclusion process on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities: (1) the probability that site x is occupied at time t; (2) a correlation…
A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is non-perturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free…
This is a joint introduction to classical and free probability, which are twin sisters. We first review the foundations of classical probability, notably with the main limiting theorems (CLT, CCLT, PLT, CPLT), and with a look into examples…