Censored Glauber Dynamics for the mean field Ising Model

We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($\beta < 1$) the mixing time is $\Theta(n\log n)$, whereas at low temperature ($\beta > 1$) it is $\exp(\Theta(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $\beta > 1$, the mixing-time of this model is $\Theta(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynamics for fixed $\beta<1$. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Currie-Weiss model. Namely, we found a scaling window of order $1/\sqrt{n}$ around the critical temperature $\beta_c=1$, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if $\beta = 1 + \delta$ for some $\delta > 0$ with $\delta^2 n \to \infty$, then the mixing-time has order $(n / \delta)\log(\delta^2 n)$. The cutoff constant is $(1/2+[2(\zeta^2 \beta / \delta - 1)]^{-1})$, where $\zeta$ is the unique positive root of $g(x)=\tanh(\beta x)-x$, and the cutoff window has order $n / \delta$.Comment: 55 pages, 4 figure

Superconducting Order Parameter in Bi-Layer Cuprates: Occurrence of π\pi Phase Shifts in Corner Junctions

We study the order parameter symmetry in bi-layer cuprates such as YBaCuO, where interesting $\pi$ phase shifts have been observed in Josephson junctions. Taking models which represent the measured spin fluctuation spectra of this cuprate, as well as more general models of Coulomb correlation effects, we classify the allowed symmetries and determine their associated physical properties. $\pi$ phase shifts are shown to be a general consequence of repulsive interactions, independent of whether a magnetic mechanism is operative. While it is known to occur in d-states, this behavior can also be associated with (orthorhombic) s-symmetry when the two sub-band gaps have opposite phase. Implications for the magnitude of $T_c$ are discussed.Comment: 5 pages, RevTeX 3.0, 9 figures (available upon request

Smoothed Complexity Theory

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

Entropy-driven cutoff phenomena

In this paper we present, in the context of Diaconis' paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature, including a chain which is non-reversible w.r.t. its stationary measure. All the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth- and-death chains this mechanism is fairly well understood; our work is an effort to generalize this picture to more general systems, such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one-dimensional problem. In those situations the drift may be looked for by means of a suitable partitioning of the state space into classes; using a statistical mechanics language it is then possible to set up a kind of energy-entropy competition between the weight and the size of the classes. Under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease.Comment: 40 pages, 1 figur

Study of the bufadienolides of the skin secretion of green toads (Bufo viridis Laur, 1758)

The composition of the skin secretion of Bufo viridis toads, which inhabit the Kharkov region, has been studied in this work

Glauber dynamics for the quantum Ising model in a transverse field on a regular tree

Motivated by a recent use of Glauber dynamics for Monte-Carlo simulations of path integral representation of quantum spin models [Krzakala, Rosso, Semerjian, and Zamponi, Phys. Rev. B (2008)], we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph $G$. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when $G$ is a regular $b$-ary tree and prove the same fast mixing results established in [Martinelli, Sinclair, and Weitz, Comm. Math. Phys. (2004)] for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the "cavity equation") together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space

Criticality in confined ionic fluids

A theory of a confined two dimensional electrolyte is presented. The positive and negative ions, interacting by a $1/r$ potential, are constrained to move on an interface separating two solvents with dielectric constants $\epsilon_1$ and $\epsilon_2$. It is shown that the Debye-H\"uckel type of theory predicts that the this 2d Coulomb fluid should undergo a phase separation into a coexisting liquid (high density) and gas (low density) phases. We argue, however, that the formation of polymer-like chains of alternating positive and negative ions can prevent this phase transition from taking place.Comment: RevTex, no figures, in press Phys. Rev.

Adiabatic times for Markov chains and applications

We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber dynamics of Ising model over Z^d/nZ^d. The theorems derived in this paper describe a type of adiabatic dynamics for l^1(R_+^n) norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with l^2(C^n) norm preserving ones, i.e. gradually changing unitary dynamics in C^n

The magnetic field influence on magnetostructural phase transition in Ni2.19Mn0.81Ga

Magnetic properties of a polycrystalline alloy Ni$_{2.19}$Mn$_{0.81}$Ga, which undergoes a first-order magnetostructural phase transition from cubic paramagnetic to tetragonal ferromagnetic phase, are studied. Hysteretic behavior of isothermal magnetization $M(H)$ has been observed in a temperature interval of the magnetostructural transition in magnetic fields from 20 to 100 kOe. Temperature dependencies of magnetization $M$, measured in magnetic fields $H = 400$ and 60 kOe, indicate that the temperature of the magnetostructural transition increases with increasing magnetic field.Comment: Presented at the Second Moscow International Symposium on Magnetism (Moscow-2002

A microscopic 2D lattice model of dimer granular compaction with friction

We study by Monte Carlo simulation the compaction dynamics of hard dimers in 2D under the action of gravity, subjected to vertical and horizontal shaking, considering also the case in which a friction force acts for horizontal displacements of the dimers. These forces are modeled by introducing effective probabilities for all kinds of moves of the particles. We analyze the dynamics for different values of the time $\tau$ during which the shaking is applied to the system and for different intensities of the forces. It turns out that the density evolution in time follows a stretched exponential behavior if $\tau$ is not very large, while a power law tail develops for larger values of $\tau$. Moreover, in the absence of friction, a critical value $\tau^*$ exists which signals the crossover between two different regimes: for $\tau < \tau^*$ the asymptotic density scales with a power law of $\tau$, while for $\tau > \tau^*$ it reaches logarithmically a maximal saturation value. Such behavior smears out when a finite friction force is present. In this situation the dynamics is slower and lower asymptotic densities are attained. In particular, for significant friction forces, the final density decreases linearly with the friction coefficient. We also compare the frictionless single tap dynamics to the sequential tapping dynamics, observing in the latter case an inverse logarithmic behavior of the density evolution, as found in the experiments.Comment: 10 pages, 15 figures, to be published in Phys. Rev.