Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values T _c < T _cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T _c or T > T _cr , three weakly periodic Gibbs measures for T = T _c , and one weakly periodic Gibbs measure for T _c < T ≤ T _cr . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 292–302, August, 2008.     

Gibbs measures for the SOS model with four states on a Cayley tree

We analyze the SOS (solid-on-solid) model with spins 0, 1, 2, 3 on a Cayley tree of order k ≥ 1. We consider translation-invariant and periodic splitting Gibbs measures for this model. The majority of the constructed Gibbs measures are mirror symmetric. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 18–31, October, 2006.     

Fertile HC models with three states on a cayley tree

We consider fertile hard-core (HC) models with three states on a homogeneous Cayley tree. It is known that four types of such models exist. For these models, we describe the translation-invariant and periodic HC Gibbs measures. We also construct a uncountable set of nonperiodic Gibbs measures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 412–424, September, 2008.     

p-adic Arakelov theory

We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. We also prove analogues of the Adjunction formula and the Riemann-Roch formula.     

Potts model with competing interactions on the Cayley tree: The contour method

We consider the Potts model with three spin values and with competing interactions of radius r = 2 on the Cayley tree of order k = 2. We completely describe the ground states of this model and use the contour method on the tree to prove that this model has three Gibbs measures at sufficiently low temperatures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 86–97, October, 2007.     

On a class of rational p-adic dynamical systems

In this paper we investigate the behavior of trajectories of one class of rational p-adic dynamical systems in complex p-adic field Cp. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the system. It is proved that such dynamical systems are not ergodic on a unit sphere with respect to the Haar measure.     

On the behavior of some two-variable p-adic L-functions

We give some p-adic integral representations for the two-variable p-adic L-functions introduced recently by G. Fox. For powers of the Teichmüller character, we use the integral representation to extend the L-function to a larger domain, in which it is a meromorphic function in the first variable and an analytic element in the second. These integral representations imply systems of congruences for the generalized Bernoulli polynomials, improving previous results of Fox, Gunaratne, and the author; they also lead to generalizations of some formulas of Diamond and of Ferrero and Greenberg for p-adic L-functions in terms of the p-adic gamma and log gamma functions.     

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.     

On uniqueness of Gibbs measure for -adic countable state Potts model on the Cayley tree

In the present paper we consider countable state of p-adic Potts model on the tree. Under some condition on weights we establish uniqueness of Gibbs measures for the model. Note that this condition does not depend on values of the prime p. An analogous fact is not true when the number of spins is finite.     

Irreducibility and p-adic monodromies on the Siegel moduli spaces

We generalize the surjectivity result of the p-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any p-rank stratum. We discuss irreducibility and connectedness of some p-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.     

Verifying a p-adic abelian Stark conjecture at s=1

In a previous paper (Ann. L’ Inst. Fourier 52(2) (2002) 379-417) the second-named author developed a new approach to the abelian p-adic Stark conjecture at s=1 and stated some related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these—the ‘Weak Refined Combined Conjecture’—in 15 cases.     

On p-adic q-l-functions and sums of powers

In this paper, we give an explicit p-adic expansion of

     

Estimates of weighted Hardy-Littlewood averages on the p-adic vector space

In the p-adic vector space , we characterize those non-negative functions ψ defined on for which the weighted Hardy-Littlewood average is bounded on (1?r?∞), and on . Also, in each case, we find the corresponding operator norm ‖Uψ‖.     

A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert-Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.     

On the purity of the free boundary condition Potts measure on random trees

We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds for the corresponding threshold value of the inverse temperature are optimal for the Ising model and differ from the Kesten Stigum bound by only 1.50% in the case q=3 and 3.65% for q=4, independently of d. Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument.     

Square integrability of representations on p-adic symmetric spaces

A symmetric space analogue of Casselman's criterion for square integrability of representations of a p-adic group is established. It is described in terms of exponents of Jacquet modules along parabolic subgroups associated to the symmetric space.