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Tracer diffusion at low temperature in kinetically constrained models

Abstract : We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient $D$ as soon as the spectral gap of the environment is positive (which coincides with the ergodicity region under general conditions). Then we study the asymptotic behaviour of $D$ when the density $1-q$ of the environment goes to 1 in two classes of KCSM. For non-cooperative models, the diffusion coefficient $D$ scales like a power of $q$, with an exponent that we compute explicitly. In the case of the Fredrickson-Andersen one-spin facilitated model, this proves a prediction made in \cite{junggarrahanchandler}. For the East model, instead we prove that the diffusion coefficient is comparable to the spectral gap, which goes to zero faster than any power of $q$. This result contradicts the prediction of physicists (\cite{junggarrahanchandler}), based on numerical simulations, that suggested $D\sim \gap^\xi$ with $\xi<1$.
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https://hal.archives-ouvertes.fr/hal-00839484
Contributor : Oriane Blondel <>
Submitted on : Friday, June 28, 2013 - 11:06:05 AM
Last modification on : Friday, March 27, 2020 - 3:50:31 AM

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Oriane Blondel. Tracer diffusion at low temperature in kinetically constrained models. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2015, 25 (3), pp.1079-1107. ⟨10.1214/14-AAP1017⟩. ⟨hal-00839484⟩

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