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SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
Abstract

Cited by 30 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
DOI: 10.1007/s0022001011611 Rigorous scaling law for the heat current in disordered harmonic chain
, 2010
"... We study the energy current in a model of heat conduction, first considered in detail by Casher and Lebowitz. The model consists of a onedimensional disordered harmonic chain of n i.i.d. random masses, connected to their nearest neighbors via identical springs, and coupled at the boundaries to Lang ..."
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We study the energy current in a model of heat conduction, first considered in detail by Casher and Lebowitz. The model consists of a onedimensional disordered harmonic chain of n i.i.d. random masses, connected to their nearest neighbors via identical springs, and coupled at the boundaries to Langevin heat baths, with respective temperatures T1 and Tn. Let EJn be the steadystate energy current across the chain, averaged over the masses. We prove that EJn ∼ (T1 −Tn)n −3/2 in the limit n → ∞, as has been conjectured by various authors over the time. The proof relies on a new explicit representation for the elements of the product of associated transfer matrices.