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

Roland Bauerschmidt, Thierry Bodineau

Research output: Contribution to journalArticlepeer-review


We present a very simple proof that the O(n) model satisfies a uniform logarithmic Sobolev inequality (LSI) if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n. This condition applies in particular to the SK spin glass model at inverse temperature β<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.

Original languageEnglish (US)
Pages (from-to)2582-2588
Number of pages7
JournalJournal of Functional Analysis
Issue number8
StatePublished - Apr 15 2019


  • Logarithmic Sobolev inequality
  • Spin glasses
  • Spin systems

ASJC Scopus subject areas

  • Analysis


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