Metastable Behavior for Bootstrap Percolation on Regular Trees

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p _T=p _T(b,θ)∈(0,1) from q _T:=q(p _T)<1 to q(p)=1 when p>p _T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p _T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h ^−1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q _T,1) converges, as h ↓0, to a well-defined measure.     

Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

This paper is concerned with the continuous-time quantum walk on ℤ, ℤ ^d , and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on ℤ, and for nearest-neighbor walks on ℤ ^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.     

Long Range Correlations and Phase Transitions in Non-equilibrium Diffusive Systems

We obtain explicit expressions for the long range correlations in the ABC model and in diffusive models conditioned to produce an atypical current of particles. In both cases, the two-point correlation functions allow one to detect the occurrence of a phase transition as they become singular when the system approaches the transition.     

A Note on Fick’s Law with Phase Transitions

Journal of Statistical Physics - We characterize the non equilibrium stationary states in two classes of systems where phase transitions are present. We prove that the interface in the limit is a...     

Connectivity Properties of Continuum Percolation Processes on ℝ2

A new approach is introduced in order to estimate the critical parameters of continuous percolation of overlapping disks in ² when the centres of the disks are Poisson distributed. Better insights on relevant parameters near criticality are found. Moreover, introducing a suitable connectivity length, the model is able to describe continuous percolation of any type of geometrical objects in ².     

Large Deviations for Subcritical Bootstrap Percolation on the Erdős–Rényi Graph

Journal of Statistical Physics - We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998)...     

Copolymers at Selective Interfaces: New Bounds on the Phase Diagram

We investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weak-coupling behavior, encoded in the slope m _c of the critical line at the origin. We focus on the directed walk case, which has turned out to be, in spite of the apparent simplicity, extremely challenging. In mathematical terms, the partition function of such a model does not depend on all the details of the Markov chain that models the polymer, but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns. This observation leads to a natural generalization of the model, in terms of arbitrary laws of return times: the most interesting case being the one of return times with power law tails (with exponent 1+α, α=1/2 in the case of the symmetric random walk). The main results we present here are:

A Geometric Analysis of the Effects of Noise on Berry Phase

In this work we describe the effect of classical and quantum noise on the Berry phase. It is not a topical review article but rather an overview of our work in this field aiming at giving a simple pictorial intuition of our results.     

Critical Exponents for the Re-entrant Phase Transitions in the Three-Dimensional Blume-Emery-Grifilths Model on the Cellular Automaton

The critical behaviour of the three-dimensional Blume-Emery-Griffiths （BEG） model ＆ investigated at D/ J = O, -0.25 and -1 in the range of-1 ≤ K/J ≤ 0 for J -- 100. The simulations are carried out on a simple cubic lattice using the heating algorithm improved from the Creutz cellular automaton （CCA ） under periodic boundary conditions. The universality of the model are obtained for re-entrant and double re-entrant phase transitions which occur at certain D/J and K/J parameters, with J and K representing the nearest-neighbour bilinear and biquadratic interactions, and D being the single-ion anisotropy parameter. The values of static critical exponents β, γ and v are estimated within the framework of the finite-size scaling theory. The results are compatible with the universal Ising critical behaviour for all continuous phase transitions in these ranges.     

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

In the present paper, we study a new kind of p-adic measures for q?+?1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.     

Research on Level Structures of Neutron-Rich Nuclei and Shape Phase Transitions Around A～100 and 150 Regions

Level structures of neutron-rich even-even nuclei around A～100 and 150 regions have been studied by observing the prompt gamma rays from the spontaneous fission of ²⁵²Cf.The highest spin is up to 10～14.The character of nuclear structures and shape phase transitions of some isotope chains are discussed.     

Geometric Phase for Two Entangled Spin-1/2 Particles in a Magnetic Field

In this paper the geometric phases of two entangled spin-1/2 particles in the presence and absence of spin-spin interaction are calculated. We also discuss the geometric phases when only one of the two particles is affected by the external magnetic field. Our results show that the geometric phase in this case is not equal to that of a single particle under the same evolution condition because of the effect of entanglement. We further study the entanglement dependence of the noncyclic geometric phases in the interacting and noninteracting spins under a time-independent uniform magnetic field. A general entanglement-dependence geometric phase is formulated.     

The Effect of Disorder on the Free-Energy for the Random Walk Pinning Model: Smoothing of the Phase Transition and Low Temperature Asymptotics

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in (Berger and Toninelli (Electron. J. Probab., to appear) and Birkner and Sun (Ann. Inst. Henri Poincaré Probab. Stat. 46:414–441, 2010; ). Given a fixed realization of a random walk Y on ℤ ^d with jump rate ρ (that plays the role of the random medium), we modify the law of a random walk X on ℤ ^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=ò₀^t 1_Xs=Ys dsL_{t}(X,Y)=\int_{0}^{t} \mathbf {1}_{X_{s}=Y_{s}}\,\mathrm {d}s: the weight of the path under the new measure is exp (βL _t (X,Y)), β∈ℝ. As β increases, the system exhibits a delocalization/localization transition: there is a critical value β _c , such that if β>β _c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d≥3, the presence of disorder makes the phase transition at least of second order. This, in dimension d≥4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.     

Terahertz subwavelength filters based on a 2D lattice of metal wires

The filtering properties of a 2D square lattice made of metallic wires are investigated through the band structures and transmission spectra by the finite-difference time-domain method. All the results show that the transmission can be affected by factors such as wave polarization, incidence direction, and wire radius. It is found that the Γ–M direction and smaller radius of metal wires are preferable for a high-frequency-pass filter for TE waves, while the Γ–X direction and comparatively greater radius of metal wires are suitable for a low-frequency-pass and a wide stop-band filter for TM waves. Band edges of the filters can be tuned by adjusting the radius of wires and lattice constant. The key features of the band structure and the corresponding transmission spectrum are strongly correlated. Our work demonstrates that detailed band maps can help understanding the transmission properties, which are essential for designing wave filters.     

Quenching effects on pulsed photoacoustic signals in NO2–air samples

A theoretical expression for the pulsed photoacoustic signal amplitude from NO₂–air mixtures is deduced based on a two-level system and the inhomogeneous wave equation, in the limit where the V-T relaxation time is longer than the laser pulse width (low buffer pressure). In this time limit, the photoacoustic signal from NO₂, at constant pressure, after excitation by pulses from a Nd:YAG laser at 532 nm, is measured at different buffer pressures. The relaxation rates are obtained by measuring the quenching of visible fluorescence induced by the same laser. The experimental dependence of the photoacoustic signal amplitude on air pressure shows a very good agreement with the model, where the measured relaxation rates are included.     

Influence of a Classical Homogeneous Gravitational Field on Dissipative Dynamics of the Phase Damped Jaynes–Cummings Model under the Markovian Approximation

In this paper, we study the dissipative dynamics of the phase damped Jaynes–Cummings model under the Markovian approximation in the presence of a classical homogeneous gravitational field. The model consists of a moving two-level atom simultaneously exposed to the gravitational field and a single-mode traveling radiation field in the presence of a phase damping mechanism. We first present the master equation for the reduced density operator of the system under the Markovian approximation in terms of a Hamiltonian describing the atom-field interaction in the presence of a homogeneous gravitational field. Then, by making use of the super-operator technique, we obtain an exact solution of the master equation. Assuming that initially the radiation field is prepared in a Glauber coherent state and the two-level atom is in the excited state, we investigate the influence of gravity on the temporal evolution of collapses and revivals of the atomic population inversion, atomic dipole squeezing, atomic momentum diffusion, photon counting statistics and quadrature squeezing of the radiation field in the presence of phase damping.     

A Note on the Extrinsic Phase Space Path Integral Method for Quantization on Riemannian Manifold Particle Motions — An Application of the Nash Embedding Theorem

We use Nash embedding for Riemann smooth manifolds to propose a constrained phase space path integral for quantization of one particle motion in a Riemannian manifold.     

Nuclear Magnetic Resonance Relaxation Study of the Phase Transitions of Rb2CuCl4·2H2O and Cs2MnCl4·2H2O Single Crystals

The physical properties and phase transitions of Rb₂CuCl₄·2H₂O and Cs₂MnCl₄·2H₂O crystals were investigated by performing ⁸⁵Rb, ⁸⁷Rb, and ¹³³Cs nuclear magnetic resonance relaxation analyses. The temperature dependences of the spectra and the spin–lattice relaxation times T ₁ near T _C are related to changes in the symmetry of the octahedrons consisting of four Cl⁻ ions and two water molecules around Rb⁺ or Cs⁺; the forms of these octahedrons are disrupted by the loss of H₂O. The difference between the relaxation times of the two crystals is possibly due to the difference between the electron structures of the Cu and Mn ions. Cu²⁺ has nine valence electrons in its 3d orbital, whereas Mn²⁺ has five valence electrons in its 3d orbital.