Gibbs measures for hardcore-solid-on-solid models on Cayley trees

Abstract We investigate the finite-state p -solid-on-solid ( p -SOS) model for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> on Cayley trees of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mn>3</mml:mn> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mn>5</mml:mn> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mn>6</mml:mn> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> . This phase diagram is qualitatively similar to the one observed for three-state p -SOS models with p &gt; 0 and, in the case of k = 2, we demonstrate that, on the level of the functional equations, the transition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> is continuous.

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