arXiv subjects
Publications and source records attributed to Sébastien Ott.
We consider finite-range ferromagnetic Ising models on $\mathbb{Z}^d$ in the regime $β<β_c$. We analyze the behavior of the prefactor to the exponential decay of $\mathrm{Cov}(σ_A,σ_B)$, for arbitrary finite sets $A$ and $B$ of even cardinality, as the distance between $A$ and $B$ diverges.
In this note we extend the analysis of a previous paper by the author to the Random Cluster model. The main result being that the pressure of the finite range ferromagnetic Ising model is analytic as a function of the inverse temperature in the regime $h=0$, $β>β_c$.
We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik.
We prove that the pressure (or free energy) of the finite range ferromagnetic Ising model on $\mathbb{Z}^d$ is analytic as a function of both the inverse temperature $β$ and the magnetic field $h$ whenever the model has the exponential weak mixing property. We also prove the exponential weak mixing property whenever $h\neq 0$. Together with known results on the regime $h=0,β<β_c$, this implies both analyticity and weak mixing in all the domain of $(β,h)$ outside of the transition line $[β_c,\infty)\times \{0\}$. The proof of analyticity uses a graphical representation of the Glauber dynamic due to Schonmann and cluster expansion. The proof of weak mixing uses the random cluster representation.
We provide a detailed analysis of the correlation length in the direction parallel to a line of modified coupling constants in the ferromagnetic Potts model on $\mathbb{Z}^d$ at temperatures $T>T_c$. We also describe how a line of weakened bonds pins the interface of the Potts model on $\mathbb{Z}^2$ below its critical temperature. These results are obtained by extending the analysis by Friedli, Ioffe and Velenik from Bernoulli percolation to FK-percolation of arbitrary parameter $q>1$.