Publications (56) View all
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Article: Finite size corrections to disordered systems on Erd\"{o}s-R\'enyi random graphs
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ABSTRACT: We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the O(1/N) corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulae on the Random Field Ising Model at zero temperature, by computing finite size corrections to the ground state energy density05/2013; -
Article: A note on weakly discontinuous dynamical transitions.
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ABSTRACT: We analyze mode coupling discontinuous transition in the limit of vanishing discontinuity, approaching the so called "A(3)" point. In these conditions structural relaxation and fluctuations appear to have universal form independent from the details of the system. The analysis of this limiting case suggests new ways for looking at the mode coupling equations in the general case.The Journal of chemical physics 02/2013; 138(6):064504. · 3.09 Impact Factor -
Article: Replica Cluster Variational Method
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ABSTRACT: We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi’s Cluster Variational Method (CVM). Using replicas and the message-passing formulation of CVM we obtain a variational expression of the replicated free energy of a system with quenched disorder, both averaged and on a single sample, and make the hierarchical ansatz using functionals of functions of fields to represent the messages. We obtain a set of integral equations for the message functionals. The main difference with the Bethe case is that the functionals appear in the equations in implicit form and are not positive definite, thus standard iterative population dynamic algorithms cannot be used to determine them. In the simplest cases the solution could be obtained iteratively using Fourier transforms. We begin to study the method considering the plaquette approximation to the averaged free energy of the Edwards-Anderson model in the paramagnetic Replica-Symmetric phase. In two dimensions we find that the spurious spin-glass phase transition of the Bethe approximation disappears and the paramagnetic phase is stable down to zero temperature on the square lattice for different random interactions. The quantitative estimates of the free energy and of various other quantities improve those of the Bethe approximation. The plaquette approximation fails to predict a second-order spin-glass phase transition on the cubic 3D lattice but yields good results in dimension four and higher. We provide the physical interpretation of the beliefs in the replica-symmetric phase as disorder distributions of the local Hamiltonian. The messages instead do not admit such an interpretation and indeed they cannot be represented as populations in the spin-glass phase at variance with the Bethe approximation. The approach can be used in principle to study the phase diagram of a wide range of disordered systems and it is also possible that it can be used to get quantitative predictions on single samples. These further developments present however great technical challenges. KeywordsSpin glasses-Cluster variation method-Replica methodJournal of Statistical Physics 04/2012; 139(3):375-416. · 1.40 Impact Factor -
Article: Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model
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ABSTRACT: We present and solve the Replica Symmetric equations in the context of the Replica Cluster Variational Method for the 2D random bond Ising model (including the 2D Edwards-Anderson spin glass model). First we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases the spin-glass transition temperatures and the tricritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T=0, also in the presence of an external field.04/2012; -
Article: Glassy Critical Points and Random Field Ising Model
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ABSTRACT: We consider the critical properties of points of continuous glass transition as one can find in liquids in presence of constraints or in liquids in porous media. Through a one loop analysis we show that the critical Replica Field Theory describing these points can be mapped in the $\phi^4$-Random Field Ising Model. We confirm our analysis studying the finite size scaling of the $p$-spin model defined on sparse random graph, where a fraction of variables is frozen such that the phase transition is of a continuous kind.03/2012;
