Narrow Results

We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron–Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.

Ising model with competing nearest–neighbors (NN) and prolonged next–nearest–neighbors (NNN) interactions on a Cayley tree has long been studied, but there are still many problems untouched. This paper tackles new Gibbs measures of Ising–Vannimenus model with competing NN and prolonged NNN interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures (TIGMs) for the model. We show that some of the measures are extreme Gibbs distributions. In this paper, we try to determine when phase transition does occur.

In this paper, we consider the Ising-Vanniminus model on an arbitrary-order Cayley tree. We generalize the results conjectured by Akın [Chinese J. Phys.54(4), 635–649 (2016) and Int. J. Mod. Phys. B31(13), 1750093 (2017)] for an arbitrary-order Cayley tree. We establish the existence and a full classification of translation-invariant Gibbs measures (TIGMs) with a memory of length 2 associated with the model on arbitrary-order Cayley tree. We construct the recurrence equations corresponding to the generalized ANNNI model. We satisfy the Kolmogorov consistency condition. We propose a rigorous measure-theoretical approach to investigate the Gibbs measures with a memory of length 2 for the model. We explain if the number of branches of the tree does not change the number of Gibbs measures. Also, we try to determine when the phase transition does occur.

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, such that , there exists a unique symmetric TISGM μ* and there are exactly three symmetric TISGMs: (a "bottom" symmetric TISGM), (a "middle" symmetric TISGM) and (a "top" symmetric TISGM). For we also construct a continuum of distinct, symmertric SGMs which are non-TI.

Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

We investigate Gibbs measures relative to Brownian motion in the case when the interaction energy is given by a double stochastic integral. In the case when the double stochastic integral is originating from the Pauli–Fierz model in nonrelativistic quantum electrodynamics, we prove the existence of its infinite volume limit.

We present a new method to prove existence and uniform a priori estimates for Gibbs measures associated with classical particle systems in a continuum. The method is based on the choice of appropriate Lyapunov functionals and on corresponding exponential bounds for the local Gibbs specification. Extensions to infinite range and multibody interactions are included.

The statistical properties of a stochastic process may be described (1) by the expectation values of the observables, (2) by the probability distribution functions or (3) by probability measures on path space. Here an analysis of level (3) is carried out for market fluctuation processes. Gibbs measures and chains with complete connections are considered. Some other topics are also discussed, in particular the asymptotic stationarity of the processes and the behavior of statistical indicators of level (1) and (2). We end up with some remarks concerning the nature and origin of the market fluctuation process and its relation to the efficient market hypothesis.

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions, which we call irreducible countable sofic shifts. We show the variational principle for topological pressure for some sub-additive sequences with tempered variation on irreducible countable sofic shifts. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. Applications are given to some dimension problems and study of factors of (generalized) Gibbs measures on certain subshifts over countable alphabets.

We prove that the Gibbs measures $\rho$ for a class of Hamiltonian equations written as

Let Δ ⊊ V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type . Let Σ_Δ be the subshift of allowable paths in the graph of which only passes through the vertices of Δ. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on , let τ_n be the point process defined as the sum of Dirac point masses at the times k > 0, suitably rescaled, for which the first n-symbols of T^kx belong to Δ. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of φ to Σ_Δ and the parameters of the limit law are explicitly computed.

Invariant quantities of the classical motion of an ideal incompressible fluid in a two-dimensional bounded domain are used to construct a family {Π_α}_α of probability measures of the Gibbs form, which are invariant under the flow. The Gibbs exponent H is given by the renormalized energy. These measures are supported by the space of configuration Γ, i.e. the fluid vorticity is concentrated in a finite number of distinct points. Properties of a deterministic vortex dynamics having Π_α as invariant measure are investigated; in particular Markov uniqueness is proven. The classical (pre-)Dirichlet form associated to Π_α is also introduced and analyzed.