Blanc SIMI 1 - Sciences de l'information, de la matière et de l'ingénierie : Mathématiques et interactions

Critical random two-dimensional models – MAC2

Submission summary

The mathematical understanding of two-dimensional models from statistical
mechanics has undergone a profound paradigm shift with the introduction of
the SLE processes by Oded Schramm in 1999. In several cases (such as the
loop-erased random walk, the uniform spanning tree, the Gaussian free
field), a complete picture of the critical behavior is now established,
while in others (like percolation and the ferromagnetic Ising model) the
exact relationship between the discrete system and its continuous scaling
limit is more evasive, and is only rigorously known (thanks to the work of
Stanislav Smirnov and others) in a few specific cases.

Intensive work is ongoing to try to establish convergence to a scaling
limit in the general case and what is known as universality (namely, the
fact that the scaling limit of a given model does not depend on the
specifics of its microscopic definition but only on its general features -
for instance, percolation on any periodic lattice is expected to exhibit
the same critical exponents even though the value of the critical point
will of course vary with the lattice).

On the other hand, going the other way and deriving properties of a
discrete model from the knowledge of its scaling limit has also been done
in a few cases (e.g. the determination of the random walk intersection
exponents by Lawler, Schramm and Werner), but so far one can argue that
this direction is not as advanced.

Another aspect of recent (and current) research in this domain is the study
of large but finite systems close to their critical points, in what is
known as the off-critical regime, at the transition between the scale-
invariant critical behavior and the non-critical system with a finite
characteristic length. This can be traced back (in the mathematical
community) to the work of Kesten on scaling relations, which relate spatial
critical exponents at criticality to those describing the approach to the
critical temperature.

The general aim of this project is to further study the interplay between
discrete and continuous models, in as many cases as possible.

Project coordination

Vincent BEFFARA (CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE RHONE-AUVERGNE) – vince.beffara@gmail.com

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

DMA, ENS CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE ILE-DE-FRANCE SECTEUR PARIS B
CNRS - UMPA CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE RHONE-AUVERGNE

Help of the ANR 194,272 euros
Beginning and duration of the scientific project: - 48 Months

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