On the two-point function of the Potts model in the saturation regime

Abstract

We consider the Random-Cluster model on ${\mathbb{Z}}^d$ with interactions of infinite range of the form $J_x=\psi (x){\mathsf{e}}^{-\rho (x)}$ with $\rho$ a norm on ${\mathbb{Z}}^d$ and $\psi$ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of ${\beta }_{\mathrm{sat}}(s)>0$ such that the inverse correlation length in the direction $s$ is constant on $[0,{\beta }_{\mathrm{sat}}(s))$), thus removing a regularity assumption used in a previous work of ours. Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein-Zernike form) for the two-point function in the whole saturation regime $(0,{\beta }_{\mathrm{sat}}(s))$. We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in $\beta$ in the regime $({\beta }_{\mathrm{sat}}(s),{\beta }_{\mathrm{c}})$.

arXiv commentSome changes in the presentation after revision. Version accepted for publication in Communications in Mathematical Physics

References in corpus (11)

• The Random-Cluster Model
• Ornstein-Zernike Theory for the finite range Ising models above T_c
• Fluctuation theory of connectivities for subcritical random cluster models
• New critical exponent inequalities for percolation and the random cluster model
• Condensation for random variables conditioned by the value of their sum
• Potts models with a defect line
• Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field
• Asymptotics of even-even correlations in the Ising model
• Sharp asymptotics of correlation functions in the subcritical long-range random-cluster and Potts models
• Non-analyticity of the correlation length in systems with exponentially decaying interactions
• Failure of Ornstein--Zernike asymptotics for the pair correlation function at high temperature and small density

Cited by in corpus (1)

• Ornstein-Zernike behavior for Ising models with infinite-range interactions