Christophe Sabot (U. Lyon 1)

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.

Departamento de Matemáticas

Pontificia Universidad Católica de Chile (PUC-Chile)

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Santiago – Chile

(+56 2) 2354 5779

Centro de Modelamiento Matemático (CMM)

Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

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Santiago – Chile