On the Six-Vertex Model’s Free Energy

Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Tatiana Tikhonovskaia

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Δ < 1. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when a= b= 1 and c≥ 1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.

Original languageEnglish (US)
Pages (from-to)1383-1430
Number of pages48
JournalCommunications In Mathematical Physics
Issue number3
StatePublished - Nov 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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