Universality for the Distance in Finite Variance Random Graphs

We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical Erdős-Rényi graph). In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like log  _ν N, where the ν depends on the capacities and N denotes the size of the graph. In addition, the random fluctuations around this asymptotic mean log  _ν N are shown to be tight. We also consider the case where the capacities are independent copies of a positive random Λ with , for some constant c and τ>3, again resulting in graphs where the degrees have finite variance. The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of van der Hofstad et al. (Random Struct. Algorithms 27(2):76–123, 2005).     

A Replica-Coupling Approach to Disordered Pinning Models

We consider a renewal process τ = {τ ₀, τ ₁,...} on the integers, where the law of τ _i − τ _i-1 has a power-like tail P(τ _i − τ _i-1 = n) = n ^−(α+1) L(n) with α ≥ 0 and L(·) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to τ. In such generality this class of problems includes, among others, (1 + d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1 + 1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase, where τ occupies a finite fraction of to a delocalized phase, where the density of τ vanishes. In absence of disorder (i.e., if the reward is independent of n), the transition is of first order for α > 1 and of higher order for α < 1. Moreover, for α ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small (but extensive) amount of disorder is known to modify the order of transition as soon as α > 1/2 [11]. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for 0 < α < 1/2 it has been proven recently by K. Alexander [2] that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2]. Moreover, we (partially) justify a small-disorder expansion worked out in [9] for α < 1/2, showing that it provides a free energy upper bound which improves the annealed one.     

Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope >2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 313–333, 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.     

Mean-Field Behavior for Long- and Finite Range Ising Model,Percolation and Self-Avoiding Walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).      

Colored Noise Enhanced Stability in a Tumor Cell Growth System Under Immune Response

In this paper, we investigate a mathematical model for describing the growth of tumor cell under immune response, which is driven by cross-correlation between multiplicative and additive colored noises as well as the nonzero cross-correlation in between. The expression of the mean first-passage time (MFPT) is obtained by virtue of the steepest-descent approximation. It is found: (i) When the noises are negatively cross-correlated (λ<0), then the escape is faster than in the case with no correlation (λ=0); when the noises are positively cross-correlated (λ>0), then the escape is slower than in the case with no correlation. Moreover, in the case of positive cross-correlation, the escape time has a maximum for a certain intensity of one of the noises, i.e., the maximum for MFPT identifies the noise enhanced stability of the cancer state. (ii) The effect of the cross-correlation time τ ₃ on the MFPT is completely opposite for λ>0 and λ<0. (iii) The self-correlation times τ ₁ and τ ₂ of colored noises can enhance stability of the cancer state, while the immune rate β can reduce it.     

Obtaining Multimode Entangled State Representation by Generalized Radon Transformation of the Wigner Operator

In a preceding paper (Fan and Lv in J. Math. Phys. 50:102108, 2009), the phase-space integration corresponding to the straight line characteristic of two different real parameters λ,τ over the Wigner operator (i.e. the Radon transformation) leads to pure-state density operator |u〉 _{λ,τ  λ,τ} 〈u|, where |u〉 _λ,τ is just the coordinate-momentum intermediate representation. In this work we show that generalized Radon transformation of the Wigner operator yields multimode density operator of continuum variables. This provides us with a new approach for obtaining multimode entangled state representation. The Weyl ordering of the Wigner operator is used in our discussions.     

Existence of many ergodic absolutely continuous invariant measures for piecewise-expandingC 2 chaotic transformations in ℝ2 on a fixed number of partitions

Let Ω be a region in ℝⁿ and letp = P_i ) _i ^1m , be a partition ofΩ into a finite number of closed subsets having piecewise C² boundaries of finite(n - 1 )dimensional measure. Let τ:Ω→Ω be piecewise C² onP where, τ_i = τ|p_i is aC ² diffeomorphism onto its image, and expanding in the sense that there exists α > 1 such that for anyi = 1, 2,...,m ‖Dτ_i ^-1 ‖ < α^-1, where Dτ_i ^-1 is the derivative matrixτ _i ^- 1 and |‖·‖ is the Euclidean matrix norm. By means of an example, we will show that the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we will construct a piecewise expanding C² transformation on a fixed partition with a finite number of elements in ℝ², but which has an arbitrarily large number of ergodic, absolutely continuous invariant measures     

A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths

A constrained diffusive random walk of n steps in ℝ ^d and a random flight in ℝ ^d , which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, 2007; J. Theor. Probab. 20:769, 2007, and J. Stat. Phys. 131:1039, 2008). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D.     

The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Effects of Self-Correlation Time and Cross-Correlation Time of Additive and Multiplicative Colored Noises for Dynamical Properties of a Bistable System

Considering a bistable system driven by additive and multiplicative colored noises with colored cross-correlation, we obtain the analytic expressions of the stationary probability distribution P _st(x), the linear relaxation time T _c , and the correlated function C(s). The effects of the noise intensity, the self-correlation time and the cross-correlation time for the bistable system are discussed. The noise intensity D speeds up relaxation of the system from unstable points, which when D < Q, the effects are the most obvious; when D > Q, the effects are damped. The self-correlation time τ₁ and τ₂ make the stationary probability distribution of the dynamical variable x be shaper and speed up the fluctuation decay of the dynamical variable x. On the contrary, the cross-correlation time τ₃ makes the stationary probability distribution of the dynamical variable x be flatter and slows down the fluctuation decay of the dynamical variable x. The effect of the self-correlation time is more projecting than the effect of the cross-correlation time. PACS number: 05.40.−a, 02.50.−r, 05.10.Gg.     

Multifragmentation and the phase transition: A systematic study of the multifragmentation of 1A GeV Au, La and Kr

A systematic analysis of the multifragmentation (MF) in fully reconstructed events from 1A GeV Au, La and Kr collisions with C has been performed. Detailed comparisons of the various fragment properties are presented as a function of excitation energy, E*_th. The charged particle multiplicity from MF stage shows a saturation beyond E*_th ∼ 8 MeV/nucleon for Kr. The universal behavior of intermediate mass fragment yields and of the size of the largest fragment is observed only for Au and La when scaled with size of the system. The Kr data are found to lack this property. Moments of the fragment size distribution show that the Kr MF is different than the MF of Au and La. A power law behavior is observed for Au and La with exponent τ>2, while for Kr τ<2. The results are compared with the statistical multifragmentation model (SMM). A single value of all the parameters of the model fits the data for all the three systems. The breakup of Au and La is consistent with a continuous phase transition. The data indicate that both E*_th and the isotope ratio temperature T _Hc-DT decrease with increase in system size at the critical point. The breakup temperature obtained from SMM also shows the same trend as seen in data. This trend is attributed primarily to the increasing Coulomb energy with finite size effects playing a smaller role. The percolation and Ising model studies for finite size neutral matter show behavior which is opposite to the one seen in the present work. EOS Collaboration     

Potts Models in the Continuum. Uniqueness and Exponential Decay in the Restricted Ensembles

In this paper we study a continuum version of the Potts model, where particles are points in ℝ ^d , d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ ⁻¹, γ>0. In mean field, for any inverse temperature β there is a value of the chemical potential λ _β at which S+1 distinct phases coexist. We introduce a restricted ensemble for each mean field pure phase which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin-Shlosman theory (Dobrushin and Shlosman in J. Stat. Phys. 46(5–6):983–1014, 1987), we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for γ small enough which implies uniqueness and exponential decay of correlations. In a second paper (De Masi et al. in Coexistence of ordered and disordered phases in Potts models in the continuum, 2008), we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.     

Stability and Instability of the KDV Solitary Wave Under the KP-I Flow

We consider the KP-I and gKP-I equations in \mathbbR × (\mathbbR/2p\mathbbZ){{\mathbb{R}}\,\times\,({\mathbb{R}}/2\pi{\mathbb{Z}})}. We prove that the KdV soliton with subcritical speed 0 < c < c* is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig (Ann Math Stud 163:181–211, 2007). For supercritical speeds c > c*, in the spirit of the work by Duyckaerts and Merle (GAFA 18:1787–1840, 2009), we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.     

Property (RD) for Hecke Pairs

As the first step towards developing noncommutative geometry over Hecke C ^∗-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G, H) is finite, we show that the Hecke pair (G, H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant’s works in Jolissaint (J K-Theory 2:723–735, 1989; Trans Amer Math Soc 317(1):167–196, 1990) to the setting of Hecke C ^∗-algebras and show that when a Hecke pair (G, H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C ^∗-algebra. Hence they have the same K ₀-groups.     

Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary β>0, we use the orthogonal polynomials techniques developed in (Killip and Nenciu in , 2005; Killip and Nenciu in Int. Math. Res. Not. 50: 2665–2701, 2004) to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.     

Signature of chaos in power spectrum

We investigate the nature of the numerically computed power spectral densityP(f, N, τ) of a discrete (sampling time interval,τ) and finite (length,N) scalar time series extracted from a continuous time chaotic dynamical system. We highlight howP(f, N, τ) differs from the true power spectrum and from the power spectrum of a general stochastic process. Non-zeroτ leads to aliasing;P(f, N, τ) decays at high frequencies as [πτ/sinπτf]², which is an aliased form of the 1/f ² decay. This power law tail seems to be a characteristic feature of all continuous time dynamical systems, chaotic or otherwise. Also the tail vanishes in the limit ofN → ∞, implying that the true power spectral density must be band width limited. In striking contrast the power spectrum of a stochastic process is dominated by a term independent of the length of the time series at all frequencies.     

Dielectric losses in statistical mixtures

The specific features of the dielectric spectra of statistical mixtures in the form of heterogeneous systems with spherical particles chaotically arranged in the space have been investigated. The distribution function of relaxation times f(τ) has been restored. It has been established that the relaxation times are continuously distributed within a wide interval [τ₁, τ₂]. Different methods for broadening the relaxation time distribution interval and approximating the relaxation time distribution function f(τ) have been analyzed. It has been demonstrated that f(τ) is a nonmonotonic function with two maxima at the boundaries and a minimum in the vicinity of the midpoint of the interval [τ₁, τ₂]. These features of the relaxation time distribution function are responsible for the large difference between the average relaxation frequencies of the permittivity and the dielectric loss (electrical conductivity).     

Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Statistics of Advective Stretching in Three-dimensional Incompressible Flows

We present a method to quantify kinematic stretching in incompressible, unsteady, isoviscous, three-dimensional flows. We extend the method of Kellogg and Turcotte (J. Geophys. Res. 95:421–432, 1990) to compute the axial stretching/thinning experienced by infinitesimal ellipsoidal strain markers in arbitrary three-dimensional incompressible flows and discuss the differences between our method and the computation of Finite Time Lyapunov Exponent (FTLE). We use the cellular flow model developed in Solomon and Mezic (Nature 425:376–380, 2003) to study the statistics of stretching in a three-dimensional unsteady cellular flow. We find that the probability density function of the logarithm of normalised cumulative stretching (log S) for a globally chaotic flow, with spatially heterogeneous stretching behavior, is not Gaussian and that the coefficient of variation of the Gaussian distribution does not decrease with time as t^-\frac12t^{-\frac{1}{2}} . However, it is observed that stretching becomes exponential log S∼t and the probability density function of log S becomes Gaussian when the time dependence of the flow and its three-dimensionality are increased to make the stretching behaviour of the flow more spatially uniform. We term these behaviors weak and strong chaotic mixing respectively. We find that for strongly chaotic mixing, the coefficient of variation of the Gaussian distribution decreases with time as t^-\frac12t^{-\frac{1}{2}} . This behavior is consistent with a random multiplicative stretching process.     

Mesonic decays of τ− lepton: Effects of neutrino mass and mass mixing

Experimentally established mesonic decays ofτ ⁻ lepton have been reexamined with the inclusion of the effects of finite neutrino mass and the associated mass mixing in the form of Kobayashi-Maskawa mixing matrix. A comparison with the experimentally predicted decay probabilities provides limits for thev _τ mass which are finite in all decays except for the lower limit in mass mixing case of the decayτ ⁻→K*⁻ (892)+v _τ for which MeV. The large error in this value is because of (i) large errors in the experimental values of life time and branching ratio for this decay and (ii) thekm mixing used in the calculations. The ratio of parity-violating to parity-conserving terms in the differential decay probabilities of various decays differs slightly from their values corresponding to those with varishingv _τ mass.