Infinite Order Differential Operators in Spaces of Entire Functions

We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane. We obtain integral representations of some particular cases of these operators and apply these results to obtain explicit solutions to some Cauchy problems for diffusion equations with nonconstant drift term

Motif based hierarchical random graphs: structural properties and critical points of an Ising model

A class of random graphs is introduced and studied. The graphs are constructed in an algorithmic way from five motifs which were found in [Milo R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Science, 2002, 298, 824-827]. The construction scheme resembles that used in [Hinczewski M., A. Nihat Berker, Phys. Rev. E, 2006, 73, 066126], according to which the short-range bonds are non-random, whereas the long-range bonds appear independently with the same probability. A number of structural properties of the graphs have been described, among which there are degree distributions, clustering, amenability, small-world property. For one of the motifs, the critical point of the Ising model defined on the corresponding graph has been studied.Comment: 18 pages, 5 figure

Quantum effects in an anharmonic crystal

A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.Розглядається модель квантових частинок, які виконують D -вимірні коливання довкола їх положень рівноваги, що утворюють d -вимірну просту кубічну ґратку Zd . Ця модель зазнає фазового переходу, коли флуктуації зміщень частинок стають макроскопічними. Таке явище описується сприйнятливостями, залежними від мацубарівських частот ωn , n є Z . Ми доводимо дві теореми, що описують термодинамічні властивості цих сприйнятливостей. Перша теорема стверджує, що сприйнятливості з ненульовими ωn залишаються обмеженими при всіх температурах, а це означає, що макроскопічні флуктуації в даній моделі є завжди неквантові. Друга теорема дає достатню умову на те, щоб і статична сприйнятливість (яка відповідає ωn = 0 ) теж була обмеженою при всіх температурах. Ця умова включає в себе масу частинки, параметри ангармонізму та інтенсивність взаємодії. Фізичний сенс цього результату полягає в тому, що для всіх D і для всіх значень температури сильні квантові ефекти унеможливлюють виникнення критичних точок і далекого порядку. Доведення проводиться в рамках підходу, у якому сприйнятливості представляються за допомогою функціональних інтегралів. Дається короткий опис головних аспектів цього підходу

Analytic and numerical study of a hierarchical spin model

A simple hierarchical scalar spin model is studied analytically and numerically in the vicinity of its critical point. The dependence of the finite size (i.e. calculated for a large but finite number of spins) susceptibility and the location of zeros of the model partition function on the number of spins at the critical point is described analytically. It is also shown analytically that the finite size correlation length in such a model diverges at the critical point slower than it is supposed in the finite size scaling theory. Certain numerical information about the critical point and ordered phase is given. In particular, the critical temperature of the model and the critical index describing the order parameter are calculated for various values of the interaction parameter.На основі аналітичних і чисельних методів вивчається проста ієрархічна скалярна спінова модель в околі її критичної точки. Аналітично описані залежності сприйнятливості і локалізації нулів статистичної суми моделі скінченого розміру (тобто, великого, але скінченого числа спінів) від кількості спінів поблизу критичної точки. Шляхом аналітичного розрахунку, зокрема, показано, що кореляційна довжина моделі скінченого розміру розбігається в критичній точці слабше ніж це передбачалось скейлінговою теорією. Обчислена критична температура і знайдений критичний показник параметра порядку для різних значень параметра взаємодії

Hierarchical Spherical Model from a Geometric Point of View

A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distribution of the block spin variable X^{\gamma}, normalized with exponents \gamma =d+2 and \gamma =d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size L^{d} in the limit L to 1 and N to \infty . Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee--Yang zeroes. The large--$N$ limit of RG transformation with L^{d} fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe \cite{W}. Although our analysis deals only with N=\infty case, it complements various aspects of that work.Comment: 27 pages, 6 figures, submitted to Journ. Stat. Phy

Gibbs states of lattice spin systems with unbounded disorder

The Gibbs states of a spin system on the lattice Zd with pair interactions Jxyσ(x) σ(y) are studied. Here <x,y> ∈ E, i.e. x and y are neighbors in Zd. The intensities Jxy and the spins σ(x), σ(y) are arbitrarily real. To control their growth we introduce appropriate sets Jq⊂RE and Sp⊂RZd and show that, for every J = (Jxy)∈Jq: (a) the set of Gibbs states Gp(J) = {μ: solves DLR, μ(Sp) = 1} is non-void and weakly compact; (b) each μ∈Gp(J) obeys an integrability estimate, the same for all μ. Next we study the case where Jq is equipped with a norm, with the Borel σ-field B(Jq), and with a complete probability measure ν. We show that the set-valued map Jq∋J → Gp(J) has measurable selections Jq∋J → μ(J) ∈Gp(J), which are random Gibbs measures. We demonstrate that the empirical distributions N-1Σn=1NπΔn(·|J,ξ), obtained from the local conditional Gibbs measures πΔn(·|J,ξ) and from exhausting sequences of Δn⊂Zd, have ν-a.s. weak limits as N→+∞, which are random Gibbs measures. Similarly, we show the existence of the ν-a.s. weak limits of the empirical metastates N-1Σn=1NδπΔn(·|J,ξ), which are Aizenman-Wehr metastates. Finally, we demonstrate that the limiting thermodynamic pressure exists under some further conditions on ν

Euclidean Gibbs States Of Quantum Lattice Systems

An approach to the description of the Gibbs states of lattice models of interacting quantum anharmonic oscillators, based on integration in infinite dimensional spaces, is described in a systematic way. Its main feature is the representation of the local Gibbs states by means of certain probability measures (local Euclidean Gibbs measures). This makes possible to employ the machinery of conditional probability distributions, known in classical statistical physics, and to define the Gibbs state of the whole system as a solution of the equilibrium (Dobrushin-LanfordRuelle) equation. With the help of this representation the Gibbs states are extended to a certain class of unbounded multiplication operators, which includes the order parameter and the fluctuation operators describing the long range ordering and the critical point respectively. It is shown that the local Gibbs states converge, when the mass of the particle tends to infinity, to the states of the corresponding classical model. A lattice approximation technique, which allows one to prove for the local Gibbs states analogs of known correlation inequalities, is developed. As a result, certain new inequalities are derived. By means of them, a number of results describing physical properties of the model are obtained