SciPost Phys. 1, 011 (2016) ·
published 19 December 2016

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Building upon the onestep replica symmetry breaking formalism, duly
understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclideanspace logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the largedistance ("infrared", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the smalldistance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for logREMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higherorder generalizations) in terms of modelspecific contributions from UV as well as IR limits. In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given logcorrelated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$noise.
SciPost Phys. 1, 009 (2016) ·
published 27 October 2016

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We study fourpoint functions of critical percolation in two dimensions, and
more generally of the Potts model. We propose an exact ansatz for the spectrum:
an infinite, discrete and nondiagonal combination of representations of the
Virasoro algebra. Based on this ansatz, we compute fourpoint functions using a
numerical conformal bootstrap approach. The results agree with MonteCarlo
computations of connectivities of random clusters.
Giulio Biroli, Charlotte Rulquin, Gilles Tarjus, Marco Tarzia
SciPost Phys. 1, 007 (2016) ·
published 25 October 2016

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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 shortrange fluctuations associated
with the local environment on Bethe lattices and longrange fluctuations that
distinguish Euclidean from Bethe lattices with the same local environment. We
find that the phase diagram in the temperaturecoupling plane is very sensitive
to the former but, at least for the $3$dimensional (square pyramid) model,
appears qualitatively or semiquantitatively unchanged by the latter. This
surprising result suggests that the meanfield 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.