Absolute continuity of non-homogeneous Gibbs measures of the Ising model on the Cayley tree

Abstract In this paper, for the Ising model on the Cayley tree of order <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:mrow> </mml:math> , a sequence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:math> of boundary conditions is constructed depending on an initial value h which defines a Gibbs measure µ h . By investigating the dynamical behaviour of the renormalisation group map associated with the model, we prove that each measure µ h is equivalent to the disordered phase <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> </mml:math> . This result shines a new light to the question closely related to the classical result by Kakutani which asserts that any two locally-equivalent probability product measures are either equivalent or mutually-singular.

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