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Article Dans Une Revue Journal of Functional Analysis Année : 2019

A very simple proof of the LSI for high temperature spin systems

Résumé

We present a very simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality (LSI) if the positive definite coupling matrix has largest eigenvalue less than $n$. This condition applies in particular to the SK spin glass model at inverse temperature $\beta < 1/4$. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry--Emery theory for the renormalised measure.

Dates et versions

hal-01721350 , version 1 (02-03-2018)

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Roland Bauerschmidt, Thierry Bodineau. A very simple proof of the LSI for high temperature spin systems. Journal of Functional Analysis, 2019, 276 (8). ⟨hal-01721350⟩
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