Related papers: The Razumov-Stroganov conjecture: Stochastic proce…
We will discuss the link between scientific explanations and probabilities, specially in relationship with statistical mechanics and the derivation of macroscopic laws from microscopic ones.
This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov…
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…
We review a recent development at the interface between discrete mathematics on one hand and probability theory and statistics on the other, specifically the use of Markov chains and their boundary theory in connection with the asymptotics…
A prediction makes a claim about a system's future given knowledge of its past. A retrodiction makes a claim about its past given knowledge of its future. The bidirectional machine is an ambidextrous hidden Markov chain that does both…
This work develops some technology for accessing the loop expansion of the Kontsevich integral of a knot. The setting is an application of the LMO invariant to certain surgery presentations of knots by framed links in the solid torus. A…
We review some recent results on connections between Brownian motion, Whittaker functions, random matrices and representation theory.
This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including…
Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight…
We make a number of observations on Conway surreal number theory which may be useful, for further developments, in both in mathematics and theoretical physics. In particular, we argue that the concepts of surreal numbers and matroids can be…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.
Some thoughts are presented on the inter-relation between beauty and truth in science in general and theoretical physics in particular. Some conjectural procedures that can be used to create new ideas, concepts and results are illustrated…
We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes,…
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the…
We exhibit the invariance of cycle affinities in finite state Markov processes under various natural probabilistic constructions, for instance under conditioning and under a new combinatorial construction that we call ``drag and drop''. We…
We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the…
We recount recent history behind building compact models of nonlinear, complex processes and identifying their relevant macroscopic patterns or "macrostates". We give a synopsis of computational mechanics, predictive rate-distortion theory,…
The spin-statistics conection is obtained for classical point particles. The connection holds within pseudomechanics, a theory of particle motion that extends classical physics to include anticommuting Grassmann variables, and which…
Through extended consideration of two wide classes of case studies -- dilute gases and linear systems -- I explore the ways in which assumptions of probability and irreversibility occur in contemporary statistical mechanics, where the…
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise…