Absence of symmetry breaking for systems of rotors with random interactions

Abstract

We prove that Gibbs states for the Hamiltonian$H = - \sum\nolimits_{xy} {\tilde J_{xy} s_x \cdot s_y }$, with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if$\tilde J_{xy} = {{J_{xy} } \mathord{\left/ {\vphantom {{J_{xy} } {\left| {x - y} \right|^\alpha }}} \right. \kern-\nulldelimiterspace} {\left| {x - y} \right|^\alpha }}$ withJ _xy i.i.d. bounded random variables with zero average, α⩾ 1 in one dimension, and α⩾2 in two dimensions.