arXiv subjects

Franck Gabriel

Publications and source records attributed to Franck Gabriel.

22 records · Page 2Linked to original sources

arXiv2016

Planar Markovian Holonomy Fields

We study planar random holonomy fields which are processes indexed by paths on the plane which behave well under the concatenation and orientation-reversing operations on paths. We define the Planar Markovian Holonomy Fields as planar random holonomy fields which satisfy some independence and invariance by area-preserving homeomorphisms properties. We use the theory of braids in the framework of classical probabilities: for finite and infinite random sequences the notion of invariance by braids is defined and we prove a new version of the de-Finetti's Theorem. This allows us to construct a family of Planar Markovian Holonomy Fields, the Yang-Mills fields, and we prove that any regular Planar Markovian Holonomy Field is a planar Yang-Mills field. This family of planar Yang-Mills fields can be partitioned into three categories according to the degree of symmetry: we study some equivalent conditions in order to classify them. Finally, we recall the notion of Markovian Holonomy Fields and construct a bridge between the planar and non-planar theories. Using the results previously proved in the article, we compute, for any Markovian Holonomy Field, the "law" of any family of contractible loops drawn on a surface.

math-ph↗

arXiv2016

Combinatorial theory of permutation-invariant random matrices I: partitions, geometry and renormalization

In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter N. Then we emulate the theory of random matrices in a combinatorial framework: for any parameter N, we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory. A renormalization of the partition algebras allows us to consider the weak convergence as a simple convergence in a fixed space. This leads us to the definition of a deformed partition algebra for any integer parameter N and to the definition of two transforms: the cumulants transform and the exclusive moments transform. Using an improved triangular inequality for the distance defined on partitions, we prove that the deformed partition algebras, endowed with a deformation of the linear forms converge as N go to infinity. This result allows us to prove combinatorial properties about geodesics and a convergence theorem for semi-groups of functions on partitions. At the end we study a sub-algebra of functions on infinite partitions with finite support : a new addition operation and a notion of R-transform are defined. We introduce the set of multiplicative functions which becomes a Lie group for the new addition and multiplication operations. For each of them, the Lie algebra is studied. The appropriate tools are developed in order to understand the algebraic fluctuations of the moments and cumulants for converging sequences. This allows us to extend all the results we got for the zero order of fluctuations to any order.

math.CO↗

arXiv2016

A combinatorial theory of random matrices III: random walks on $\mathfrak{S}(N)$, ramified coverings and the $\mathfrak{S}(\infty)$ Yang-Mills measure

The aim of this article is to study some asymptotics of a natural model of random ramified coverings on the disk of degree $N$. We prove that the monodromy field, called also the holonomy field, converges in probability to a non-random field as $N$ goes to infinity. In order to do so, we use the fact that the monodromy field of random uniform labelled simple ramified coverings on the disk of degree $N$ has the same law as the $\mathfrak{S}(N)$-Yang-Mills measure associated with the random walk by transposition on $\mathfrak{S}(N)$. This allows us to restrict our study to random walks on $\mathfrak{S}(N)$: we prove theorems about asymptotics of random walks on $\mathfrak{S}(N)$ in a new framework based on the geometric study of partitions and the Schur-Weyl-Jones's dualities. In particular, given a sequence of conjugacy classes $(λ\_N \subset \mathfrak{S}(N))\_{N \in \mathbb{N}}$, we define a notion of convergence for $(λ\_N)\_{N \in \mathbb{N}}$ which implies the convergence in non-commutative distribution and in $\mathcal{P}$-expectation of the $λ\_N$-random walk to a $\mathcal{P}$-free multiplicative L{é}vy process. This limiting process is shown not to be a free multiplicative L{é}vy process and we compute its log-cumulant functional. We give also a criterion on $(λ\_N)\_{N \in \mathbb{N}}$ in order to know if the limit is random or not.

math.PR↗

arXiv2016

The generalized master fields

The master field is the large $N$ limit of the Yang-Mills measure on the Euclidean plane. It can be viewed as a non-commutative process indexed by paths on the plane. We construct and study generalized master fields, called free planar Markovian holonomy fields, which are versions of the master field where the law of a simple loop can be as more general as it is possible. We prove that those free planar Markovian holonomy fields can be seen as well as the large $N$ limit of some Markovian holonomy fields on the plane with unitary structure group.

math-ph↗