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Register a new userCorrelation decay for hard spheres via Markov chains
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Added by
Will Perkins
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of $2$ and $1.7$, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results.
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We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained in cite{BFG}. By improving such results we also show, under additional assumptions, that the LDP holds with the strong L^1 topology on the space of currents. We deduce a general version of the Gallavotti-Cohen (GC) symmetry for the current field and show that it implies the so-called fluctuation theorem for the GC functional. We also analyze the large deviation properties of generalized empirical currents associated to a fundamental basis in the cycle space, which, as we show, are given by the first class homological coefficients in the graph underlying the Markov chain. Finally, we discuss in detail some examples.
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We review recent results on the metastable behavior of continuous-time Markov chains derived through the characterization of Markov chains as unique solutions of martingale problems.
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