GRADIENT GIBBS MEASURES FOR THE SOS MODEL WITH COUNTABLE VALUES ON A CAYLEY TREE

F Henning (1) , C Kuelske (1) , A Le Ny (2, 3, 4) , U A Rozikov (5)

1 RUB - Ruhr University Bochum = Ruhr-Universität Bochum
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
4 PRES Université Paris-Est
5 Institute of Mathematics [Tashkent]

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k ≥ 2 and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some two periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k ≥ 2, which define GGMs different from the 4-periodic ones. Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (sec-ondary)

Mots clés: SOS model; Cayley tree; Gibbs measures; tree-indexed Markov chain; gradient Gibbs measures; boundary law

Domaines: Probabilités [math.PR]

F Henning, C Kuelske, A Le Ny, U A Rozikov. GRADIENT GIBBS MEASURES FOR THE SOS MODEL WITH COUNTABLE VALUES ON A CAYLEY TREE. 2019. ⟨hal-02037877⟩