TY - JOUR

T1 - On the Six-Vertex Model’s Free Energy

AU - Duminil-Copin, Hugo

AU - Kozlowski, Karol Kajetan

AU - Krachun, Dmitry

AU - Manolescu, Ioan

AU - Tikhonovskaia, Tatiana

N1 - Publisher Copyright:
© 2022, The Author(s).

PY - 2022/11

Y1 - 2022/11

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-022-04459-x

DO - 10.1007/s00220-022-04459-x

M3 - Article

C2 - 36263094

AN - SCOPUS:85137778033

SN - 0010-3616

VL - 395

SP - 1383

EP - 1430

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 3

ER -