Alexei Daletskii

A Phase Transition in a Quenched Amorphous Ferromagnet

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over

Laplace operators in deRham complexes associated with measures on configuration spaces

\mathbb {R}^d

Equilibrium stochastic dynamics of Poisson cluster ensembles

Rd,

Stochastic equations and Dirichlet operators on infinite product manifolds

d\ge 2

Stochastic Analysis on (Infinite-Dimensional) Product Manifolds

d≥2. Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the intensity of the underlying point process and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.

STOCHASTIC EQUATIONS AND QUASI-INVARIANCE ON INFINITE PRODUCT GROUPS

The distribution µ of a Gibbs cluster point process in χ = ℝd (with n-point clusters) is studied via the projection of an auxiliary Gibbs measure defined on the space of configurations in χ × χn. We show that µ is quasi-invariant with respect to the group Diff0(χ) of compactly supported diffeomorphisms of χ and prove an integration-by-parts formula for µ. The corresponding equilibrium stochastic dynamics is then constructed by using the method of Dirichlet forms.