11,448 research outputs found
Stochastic domination for the Ising and fuzzy Potts models
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 , \Td. For given interaction parameters ,
and external field h_1\in\RR, we compute the smallest external field
such that the plus measure with parameters and dominates
the plus measure with parameters and for all .
Moreover, we discuss continuity of with respect to the three
parameters , , 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.Comment: 22 pages, 5 figure
Smoothing and filtering with a class of outer measures
Filtering and smoothing with a generalised representation of uncertainty is
considered. Here, uncertainty is represented using a class of outer measures.
It is shown how this representation of uncertainty can be propagated using
outer-measure-type versions of Markov kernels and generalised Bayesian-like
update equations. This leads to a system of generalised smoothing and filtering
equations where integrals are replaced by supremums and probability density
functions are replaced by positive functions with supremum equal to one.
Interestingly, these equations retain most of the structure found in the
classical Bayesian filtering framework. It is additionally shown that the
Kalman filter recursion can be recovered from weaker assumptions on the
available information on the corresponding hidden Markov model
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