Title: Boltzmann triangulations with Ising model on faces
Abstract:
We consider the Boltzmann random triangulation of a polygon coupled with
Ising model on its faces. In a previous work, Bernardi and Bousquet-Melou
studied an equivalent model under monochromatic boundary conditions for small
boundary length, identifying a unique critical point of the model.
We generalize their results to Ising-coupled triangulations with Dobrushin
boundary conditions and arbitrary boundary length. Using explicit expression
of the partition function at the critical point we show that the perimeter
exponent of the model is 7/3 instead of 5/2 for uniform triangulations.
We also show that any Ising interface only touches the boundary a finite
number of times as the boundary size tends to infinity.
Time permitting, we also discuss ideas on how to construct an infinite
Ising triangulation of the half plane with Dobrushin boundary conditions,
which we conjecture to be the local limit of finite Boltzmann Ising
triangulations with a Dobrushin boundary in the sense of Benjamini-Schramm
as the perimeter goes to infinity. This is a joint ongoing work with
Linxiao Chen (University of Paris-Sud).