Senya Shlosman

Interfaces in the Potts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation

We develop a new analysis of the order-disorder transition in ferromagnetic Potts models for large numberq of spin states. We use the Pirogov-Sinaï theory which we adapt to the Fortuin-Kasteleyn representation of the models. This theory applies in a rather direct way in our approach and leads to a system of non-interacting contours with small activities. As a consequence, simpler and more natural techniques are found, allowing us to recover previous results on the bulk properties of the model (which then extend to non-integer values ofq) and to deal with non-translation invariant boundary conditions. This will be applied in a second part of this work to study the behaviour of the interfaces at the transition point.

Complete analyticity for 2D Ising completed

We study the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh. We extend to every subcritical value of the temperature a result previously proven by Martirosyan at low enough temperature, and which roughly states that for finite systems with — boundary conditions under a positive external field, the boundary effect dominates in the bulk if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the bulk. As a consequence we are able to complete the proof that “complete analyticity for nice sets” holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model.The main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, and recently extended to all temperatures below the critical one by Ioffe.

Constrained Variational Problem with Applications to the Ising Model

We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with (−)-boundary conditions under a positive external field, the boundary effect dominates in the system if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the system. In our earlier work this result was extended to every subcritical value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical valueB0(T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a “squeezed Wulff shape”. The main step in our study is the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.

The droplet in the tube: a case of phase transition in the canonical ensemble

AbstractWe consider the 2-dimensional Ising model with ferromagnetic nearest neighbour interaction at inverse temperatureβ. LetSN=Σσt be the total magnetization inside anN×N square boxΛ,μΛper be the Gibbs state inΛ with periodic b.c., andm(β) be the spontaneous magnetization. We show the existence of the limit

Ergodicity of probabilistic cellular automata: a constructive criterion

\psi (\varrho ) = \mathop {\lim }\limits_{N \to \infty } \left( { - \frac{1}{{\beta N}}} \right)\ln \mu _\Lambda ^{per} (S_N = [N\varrho ])

Interfaces in the Potts Model II: Antonov's Rule and Rigidity of the Order Disorder Interface

for |ϱ|<m(β), providedβ is large enough. It turns out that the quantityψ(ϱ) is closely related to the Wulf construction, and the dependence of the functionψ(ϱ) onϱ is singular.

Aggregation and intermediate phases in dilute spin systems

We consider various sufficiently nonlinear vector models of ferromagnets, of nematic liquid crystals and of nonlinear lattice gauge theories with continuous symmetries. We show, employing the method of Reflection Positivity and Chessboard Estimates, that they all exhibit first-order transitions in the temperature, when the nonlinearity parameter is large enough. The results hold in dimension 2 or more for the ferromagnetic models and the RPN−1 liquid crystal models and in dimension 3 or more for the lattice gauge models. In the two-dimensional case our results clarify and solve a recent controversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.

A Manifold of Pure Gibbs States of the Ising Model on a Cayley Tree

We give a sequence of criteria (of increasing complexity) for the exponential ergodicity of discrete time interacting particle systems. Each criterion involves estimating the dependence on initial conditions of the process on finite space-time volumes. It generalizes and improves the existing single site condition and is the analog of the Dobrushin-ShlosmanCv condition in equilibrium statistical mechanics. Our “dynamic” criterion may also be used to prove the uniqueness of Gibbs state in situations where theCv condition fails. As a converse we prove that if there is a certain form of convergence to the stationary measure faster thann−d, wheren is the time andd is the dimension of the lattice, then our condition holds for some space-time volumes and hence the convergence must be exponentially fast.