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Abstract

In this thesis we consider two problems related to the multidimensional East model on \(\mathbb{Z}^d\), a well studied kinetically constrained model (KCM). KCM are interacting particle models in which the local configurations are updated with equilibrium only if the configuration in the neighbourhood of the update satisfies certain constraints. Usually KCM are defined on a local \(0\)-\(1\)-state space where the \(0\)s (or \emph{vacancies}) are the facilitating states and the \(1\)s (or \emph{particles}) are the neutral ones. For the East model the constraints for the update at \(x\in \mathbb{Z}^d\) require a vacancy on a smaller neighbour \(y\) in the lexicographic order.

The first problem is a front evolution problem as the equilibrium density \(q\) of the facilitating vertices vanishes. Starting with a unique unconstrained vertex at the origin, let \(C(t)\) consist of those vertices which became unconstrained within time \(t\) and, for an arbitrary positive direction \(\mathbf{x}\in \mathbb{R}^d_+\), let \(v_{\max}(\mathbf{x}), v_{\min}(\mathbf{x})\) be the maximal/minimal velocities at which \(C(t)\) grows in that direction. If \(\mathbf{x}\) is independent of \(q\), we prove that \(v_{\max}(\mathbf{x})={v_{\min}(\mathbf{x})}^{(1+o(1))}=\gamma_d^{(1+o(1))}\) as \(q\rightarrow 0\), where \(\gamma_d\) is the spectral gap of the process on \(\mathbb{Z}^d\). We also analyse the case in which some of the coordinates of \(\mathbf{x}\) vanish as \(q\rightarrow 0\). In particular, for \(d=2\) we prove that if \(\mathbf{x}\) approaches one of the two coordinate directions fast enough, then \(v_{\max}(\mathbf{x})={v_{\min}(\mathbf{x})}^{(1+o(1))}=\gamma_1^{(1+o(1))}=\gamma_d^{d(1+o(1))}\), i.e.\ the growth of \(C(t)\) close to the coordinate directions is dictated by the one-dimensional process. As a result the region \(C(t)\) becomes extremely elongated inside \(\mathbb{Z}^d_+\). Using these bounds on the front speed we identify an elongated subset \(S(t)\subset C(t)\) that grows in \(t\) and which is mixing in \(t\rightarrow \infty\). In fact, remarkably, these bounds on the front speed together with past results also imply a cutoff result for the East process on a box in \(\mathbb{Z}^d\).

The second problem is a coarse-grained model of glass forming liquids introduced by Chandler and Garrahan which is closely related to the East model. Instead of fixing a facilitation direction in the model, we consider multiple types of facilitating vertices that evolve on the same lattice, where each type behaves like a rotated version of the East process, e.g.\ in \(d=2\) one type requires a vacancy in the south-west neighbourhood of updating vertices, one south-east, one north-east and one north-west. The crux is that the neutral vertices, i.e.\ the particles in the East model, are shared for all types of facilitating vertex. We call this model the \emph{multicolour East model} (MCEM). We prove that if the number of species is equal to the maximum amount of possible rotations the associated process, the MCEM process, is not ergodic. We then provide sufficient conditions so that the MCEM process is ergodic and the spectral gap positive in \(\mathbb{Z}^d\). For example we show that in \(d=2\) any MCEM process with three types of facilitating vertices has a positive spectral gap. Further, for \(d=2\), we analyse the scaling of the spectral gap when the minimum density \(q_{\min}\) of the facilitating vertex types tends to zero. We show sufficient conditions on the equilibrium distribution of the vertex types that the spectral gap tends to \(\gamma_2(q_{\min})\) as \(q_{\min}\rightarrow 0\). In particular, we show that this is also the case when vertices of the least frequent facilitating vertex type are surrounded by vertices of different types that inhibit their movement. We do this through a fine analysis, whereby the frequent vertex type move and remove each other in such a way as to clear the way for an effective two-dimensional motion of the infrequent types.

A novel technical ingredient is a detailed analysis of the asymptotics of a principal Dirichlet eigenvalue based on the renormalisation technique of~Chleboun, Faggionato and Martinelli. This analysis enters in both sets of results.

Recommended citation: Y. Couzinié. “The multidimensional East model: a multicolour model and a front evolution problem”. Ph.D thesis, Roma Tre University, 2022.

BibTeX (also as a download):

@phdthesis{couziniePhd,
        author = "Yannick Couzini\'{e}",
        title = "The multidimensional East model: a multicolour model and a front evolution problem",
        school = "Roma Tre University",
        year = "2022",
        address = "Rome, Italy",
        month = jun
}