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Louis-Pierre Arguin 《Journal of statistical physics》2007,126(4-5):951-976
We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's
Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used
to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection
between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we
rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure. 相似文献
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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. 相似文献
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