Propagation of Chaos and Residual Dependence in Gibbs Measures on Finite Sets

Abstract We compare a mean-field Gibbs distribution on a finite state space on N spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni [3], where marginal distributions of size $$k=o(N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are compared to product measures.