Related papers: (Co)monads in Free Probability Theory
We review some important algebraic structures which appear in a priori remote areas of Mathematics, such as control theory, numerical methods for solving differential equations, and renormalization in Quantum Field Theory. Starting with…
In this paper, we develop the notion of free-Boolean independence in an amalgamation setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence…
We consider gauge theories from the free evolution point of view, in which initial data satisfying constraints of a theory are given. Because the constraints are compatible with the field equations they remain so. We study a model…
In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and statistics. In this paper, we connect some ideas from…
We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and…
We present a new description of the known large deviation function of the classical symmetric simple exclusion process by exploiting its connection with the quantum symmetric simple exclusion processes and using tools from free probability.…
In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie…
The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf…
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.
We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of…
We construct pairs of algebras with mixed independence relations by using truncations of reduced free products of algebras. For example, we construct free-Boolean pairs of algebras and free-monotone pairs of algebras. We also introduce…
Based on the monoid classifier, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and…
Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical…
After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic…
We study the problem of conditional expectations in free random variables and provide closed formulas for the conditional expectation of resolvents of arbitrary non-commutative polynomials in free random variables onto the subalgebra of an…
Recent works have shown that defining a behavioural equivalence that matches the observational properties of a quantum-capable, concurrent, non-deterministic system is a surprisingly difficult task. We explore coalgebras over distributions…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
In this paper, we will consider R-transform theory and R-transform calculus for compatible noncommutative probability space and amagamated noncommutative probability space. By doing this, we can realize the relation between scalar-valued…
The most standard description of symmetries of a mathematical structure produces a group. However, when the definition of this structure is motivated by physics, or information theory, etc., the respective symmetry objects might become more…