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Stochastic domination for the Ising and fuzzy Potts models

Marcus Warfheimer (Institutionen för matematiska vetenskaper, matematik)

We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\Td$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\in\RR$, we compute the smallest external field $\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\geq\tilde{h}$. Moreover, we discuss continuity of $\tilde{h}$ with respect to the three parameters $J_1$, $J_2$, $h$ and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on $\Zd$ the fuzzy Potts measures dominate the same set of product measures while on $\Td$, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures. For the Ising model, Liggett and Steif proved that on $\Zd$ the plus measures dominate the same set of product measures while on $\T^2$ that statement fails completely except when there is a unique phase.

Denna post skapades 2010-03-19. Senast ändrad 2010-04-20.
CPL Pubid: 118145


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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)


Annan matematik

Chalmers infrastruktur

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Denna publikation ingår i:

Interacting particle systems in varying environment, stochastic domination in statistical mechanics and optimal pairs trading in finance

Ingår i serie

Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University 2010:21