We will give an overview on recent developpements on two self-interacting processes : the Edge Reinforced Random Walk (ERRW)

and the Vertex Reinforced Jump Process (VRJP).

The ERRW and VRJP are known since a few years to be related to a supersymetric field

considered by Disertori, Spencer and Zirnbauer.

On finite graphs we introduce a random Schrödinger operator with a

random potential, with decorrelation at distance 2, from which the

mixing field of the VRJP can be described from the Green function.

The distribution of this potential is explicite and appears to be new,

and can be understood as a multivariate generalization of the inverse

gaussian law.

Interesting phenomenons appear by extending this representation to

infinite graphs.

In particular, the transience of the VRJP is signed by the existence of

a positive generalized eigenfunction of the random Schrödinger operator.

The VRJP can then be represented as a mixture of Markov Jump processes

with a field involving the Green function of the random Schrödinger

operator, the generalized eigenfunction and an extra independent random

variable that governs the escape probability.

From this we can infer a functional CLT at weak disorder for the VRJP

and ERRW in dimension d>2.

Based on joint works with, P. Tarrès, X. Zeng.

Pontificia Universidad Católica de Chile (PUC-Chile)

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Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

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Santiago – Chile