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 $\mathbb{R}^d$, $d\geq 2$. 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 particle density (i.e., 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 dynamics of particle systems on unbounded degree graphs

We consider an infinite system of coupled stochastic differential equations (SDE) describing dynamics of the following infinite particle system. Each partricle is characterised by its position $x\in \mathbb{R}^{d}$ and internal parameter (spin) $\sigma _{x}\in \mathbb{R}$. While the positions of particles form a fixed ("quenched") locally-finite set (configuration) $\gamma \subset$ $\mathbb{R}^{d}$, the spins $\sigma _{x}$ and $\sigma _{y}$ interact via a pair potential whenever $\left\vert x-y\right\vert 0$ is a fixed interaction radius. The number $n_{x}$ of particles interacting with a particle in positionn $x$ is finite but unbounded in $x$. The growth of $n_{x}$ as $x\rightarrow \infty$ creates a major technical problem for solving our SDE system. To overcome this problem, we use a finite volume approximation combined with a version of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a scale of Banach spaces of weighted sequences. As an application example, we construct stochastic dynamics associated with Gibbs states of our particle system

Stochastic differential equations in a scale of Hilbert spaces

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in R

Gibbs states of continuum particle systems with unbounded spins : existence and uniqueness

We study an infinite system of particles chaotically distributed over a Euclidean space Rd. Particles are characterized by their positions x∈Rd and an internal parameter (spin) σx∈Rm and interact via position-position and (position dependent) spin-spin pair potentials. Equilibrium states of such system are described by Gibbs measures on a marked configuration space. Due to the presence of unbounded spins, the model does not fit the classical (super-) stability theory of Ruelle. The main result of the paper is the derivation of sufficient conditions of the existence and uniqueness of the corresponding Gibbs measures