# Role of fluctuations in the phase transitions of coupled plaquette spin models of glasses

### Submission summary

 As Contributors: Charlotte Rulquin Arxiv Link: http://arxiv.org/abs/1606.08268v3 Date accepted: 2016-10-19 Date submitted: 2016-10-14 Submitted by: Rulquin, Charlotte Submitted to: SciPost Physics Discipline: Physics Subject area: Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

We study the role of fluctuations on the thermodynamic glassy properties of plaquette spin models, more specifically on the transition involving an overlap order parameter in the presence of an attractive coupling between different replicas of the system. We consider both short-range fluctuations associated with the local environment on Bethe lattices and long-range fluctuations that distinguish Euclidean from Bethe lattices with the same local environment. We find that the phase diagram in the temperature-coupling plane is very sensitive to the former but, at least for the $3$-dimensional (square pyramid) model, appears qualitatively or semi-quantitatively unchanged by the latter. This surprising result suggests that the mean-field theory of glasses provides a reasonable account of the glassy thermodynamics of models otherwise described in terms of the kinetically constrained motion of localized defects and taken as a paradigm for the theory of dynamic facilitation. We discuss the possible implications for the dynamical behavior.

### Ontology / Topics

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Published as SciPost Phys. 1, 007 (2016)

In this new version we implemented the modifications asked by the referee. On the 5 points raised by her/him, 4 resulted in revisions in the main text of the manuscript. For the last one, we gave an answer to the referee on the submission page.

### List of changes

Concerning the points raised by the referee, the following changes have been made:

1. Section 2.2, equation (5) : we changed $\beta \epsilon$ in $- \beta \epsilon$.

2. Revisions have been implemented in the main text (p.7, section 3.1) on the boundary conditions allowing the one-to-one mapping, and on the consequence for $c$ plaquettes to share the same spin.

"As shown for instance in Refs. \cite{newman99,garrahan00,ritort-sollich,garrahan_annealed}, the mapping from one representation to the other is one-to-one with periodic boundary conditions (at least for the TPM and SPyM studied here). The correspondence is not exactly one-to-one for others boundary conditions, but is recovered in the thermodynamic limit.

In terms of the plaquette variables, one can reexpress the Hamiltonian in Eq. (???) as
[...]
which corresponds to a noninteracting Ising model in an external field $J/2$. As is well-known\cite{newman99, garrahan00, garrahan02,ritort-sollich}, the dynamics is nonetheless glassy and the single-spin flip dynamics maps onto a relaxation with kinetic constraints for the plaquette variables. The fact that $c$ plaquettes are connected to one and the same spin leads to this nontrivial dynamics."

3. Soon after equation (14), the squared mean is subtracted in order to recover the variance.

4. A sentence is added at the end of section 3.1.2.
"To summarize, self-duality holds only in the annealed case for c=p.”

### Submission & Refereeing History

Resubmission 1606.08268v3 on 14 October 2016
Submission 1606.08268v1 on 28 July 2016