TY - JOUR

T1 - Loss without recovery of Gibbsianness during diffusion of continuous spins

AU - Külske, C.

AU - Redig, F.H.J.

PY - 2006

Y1 - 2006

N2 - We consider a specific continuous-spin Gibbs distribution µt=0 for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions.
For `high temperature' initial measures we prove that the time-evoved measure µt is Gibbsian for all t. For `low temperature' initial measures we prove that µt stays Gibbsian for small enough times t, but loses its Gibbsian character for large enough t. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large t in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension d=2. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.

AB - We consider a specific continuous-spin Gibbs distribution µt=0 for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions.
For `high temperature' initial measures we prove that the time-evoved measure µt is Gibbsian for all t. For `low temperature' initial measures we prove that µt stays Gibbsian for small enough times t, but loses its Gibbsian character for large enough t. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large t in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension d=2. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.

U2 - 10.1007/s00440-005-0469-9

DO - 10.1007/s00440-005-0469-9

M3 - Article

SN - 0178-8051

VL - 135

SP - 428

EP - 456

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3

ER -