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18
A connection between the GhirlandaGuerra identities and ultrametricity.
, 2009
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PoissonDirichlet statistics for the extremes of a logcorrelated Gaussian field, preprint, arXiv:1203.4216 [math.PR
, 2012
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The Generalized Random Energy Model and its Application to the Statistical Physics of Ensembles of Hierarchical Codes
, 2007
"... In an earlier work, the statistical physics associated with finite–temperature decoding of code ensembles, along with the relation to their random coding error exponents, were explored in a framework that is analogous to Derrida’s random energy model (REM) of spin glasses, according to which the ene ..."
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In an earlier work, the statistical physics associated with finite–temperature decoding of code ensembles, along with the relation to their random coding error exponents, were explored in a framework that is analogous to Derrida’s random energy model (REM) of spin glasses, according to which the energy levels of the various spin configurations are independent random variables. The generalized REM (GREM) extends the REM in that it introduces correlations between energy levels in an hierarchical structure. In this paper, we explore some analogies between the behavior of the GREM and that of code ensembles which have parallel hierarchical structures. In particular, in analogy to the fact that the GREM may have different types of phase transition effects, depending on the parameters of the model, then the above–mentioned hierarchical code ensembles behave substantially differently in the various domains of the design parameters of these codes. We make an attempt to explore the insights that can be imported from the statistical mechanics of the GREM and be harnessed to serve for code design considerations and guidelines.
Local energy statistics in spin glasse
 J. Stat. Phys
"... Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. We review some rigorous results confirming the validity of this conjecture. In the context of the SK m ..."
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Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. We review some rigorous results confirming the validity of this conjecture. In the context of the SK models, we analyse the limits of the validity of the conjecture for energy levels growing with the volume of the system. In the case of the Generalised Random energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies EN < βcN, where βc is the critical temperature. We also explain the more complex behaviour that sets in at higher energies. 1
Variational Bounds for the Generalized Random Energy Model
, 2006
"... We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds”, and identify the random probability cascade as the appropriate random ..."
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Cited by 3 (2 self)
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We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds”, and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand’s concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.
Generalized Random Energy Model at complex temperatures
, 2014
"... Abstract. Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature β. We compute the limiting logpartition function and describe the fluctuations of the partition function. For the GREM with ..."
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Abstract. Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature β. We compute the limiting logpartition function and describe the fluctuations of the partition function. For the GREM with d levels, in total, there are 1 2 (d+ 1)(d+ 2) phases, each of which can symbolically be encoded as Gd1F d2Ed3 with d1, d2, d3 ∈ N0 such that d1 + d2 + d3 = d. In phase Gd1F d2Ed3, the first d1 levels (counting from the root of the GREM tree) are in the glassy phase (G), the next d2 levels are dominated by fluctuations (F), and the last d3 levels are dominated by the expectation (E). Only the phases of the form Gd1Ed3 intersect the real β axis. We describe the limiting distribution of the zeros of the partition function in the complex β plane ( = Fisher zeros). It turns out that the complex zeros densely touch the positive real axis at d points at which the GREM is known to undergo phase transitions. Our results confirm rigorously and considerably extend the replicamethod predictions from the physics literature. Figure 1. Phase diagram of the GREM in the complex β plane together with the level lines of the limiting logpartition function. See Figure 4 for details.
Fluctuations of the partition function in the GREM with external field
, 2008
"... We study Derrida’s generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchica ..."
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We study Derrida’s generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarsegraining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarsegrained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by strength of external field and by parameters of the GREM.
POISSONDIRICHLET STATISTICS FOR THE EXTREMES OF THE TWODIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
, 2013
"... Abstract. In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a logcorrelated Gaussian field converge to a PoissonDirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on sh ..."
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Abstract. In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a logcorrelated Gaussian field converge to a PoissonDirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a onestep replica symmetry breaking in spin glass terminology. This implies PoissonDirichlet statistics by general spin glass arguments. In this note, this approach is used to prove PoissonDirichlet statistics for the twodimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis. 1.
Mathematical Physics Local Energy Statistics in Disordered Systems: A Proof of the Local REM Conjecture
, 2006
"... Abstract: Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to be sat ..."
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Abstract: Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to be satisfied in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. 1.