## The Annals of Applied Probability

### Tracer diffusion at low temperature in kinetically constrained models

Oriane Blondel

#### 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 behavior of $D$ when the density $1-q$ of the environment goes to $1$ in two classes of KCSM. For noncooperative 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 Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205]. 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 (Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205; J. Chem. Phys. 123 (2005) 084509]), based on numerical simulations, that suggested $D\sim\operatorname{gap}^{\xi}$ with $\xi<1$.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1079-1107.

Dates
First available in Project Euclid: 23 March 2015

https://projecteuclid.org/euclid.aoap/1427124124

Digital Object Identifier
doi:10.1214/14-AAP1017

Mathematical Reviews number (MathSciNet)
MR3325269

Zentralblatt MATH identifier
1317.82054

#### Citation

Blondel, Oriane. Tracer diffusion at low temperature in kinetically constrained models. Ann. Appl. Probab. 25 (2015), no. 3, 1079--1107. doi:10.1214/14-AAP1017. https://projecteuclid.org/euclid.aoap/1427124124

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