Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network

We consider the ferromagnetic Ising model on a highly inhomogeneous network created by a growth process. We find that the phase transition in this system is characterised by the Berezinskii--Kosterlitz--Thouless singularity, although critical fluctuations are absent, and the mean-field description is exact. Below this infinite order transition, the magnetization behaves as $exp(-const/\sqrt{T_c-T})$. We show that the critical point separates the phase with the power-law distribution of the linear response to a local field and the phase where this distribution rapidly decreases. We suggest that this phase transition occurs in a wide range of cooperative models with a strong infinite-range inhomogeneity. {\em Note added}.--After this paper had been published, we have learnt that the infinite order phase transition in the effective model we arrived at was discovered by O. Costin, R.D. Costin and C.P. Grunfeld in 1990. This phase transition was considered in the papers: [1] O. Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531 (1990); [2] O. Costin and R.D. Costin, J. Stat. Phys. 64, 193 (1991); [3] M. Bundaru and C.P. Grunfeld, J. Phys. A 32, 875 (1999); [4] S. Romano, Mod. Phys. Lett. B 9, 1447 (1995). We would like to note that Costin, Costin and Grunfeld treated this model as a one-dimensional inhomogeneous system. We have arrived at the same model as a one-replica ansatz for a random growing network where expected to find a phase transition of this sort based on earlier results for random networks (see the text). We have also obtained the distribution of the linear response to a local field, which characterises correlations in this system. We thank O. Costin and S. Romano for indicating these publications of 90s.Comment: 5 pages, 2 figures. We have added a note indicating that the infinite order phase transition in the effective model we arrived at was discovered in the work: O. Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531 (1990). Appropriate references to the papers of 90s have been adde

Markov vs. nonMarkovian processes A comment on the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T. D. Frank. Our comment centers on the claims of a nonlinear Markov process and a nonlinear Fokker-Planck equation. First, memory in transition densities is misidentified as a Markov process. Second, Frank assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was given that a Chapman-Kolmogorov equation exists for memory-dependent processes. A nonlinear Markov process is claimed on the basis of a nonlinear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the transition probabilities, is either an ordinary linearly generated Markovian one, or else is a linearly generated nonMarkovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a nonlinear Markov process nor nonlinear Fokker-Planck equation for a transition density. The confusion rampant in the literature arises in part from labeling a nonlinear diffusion equation for a 1-point probability density as nonlinear Fokker-Planck, whereas neither a 1-point density nor an equation of motion for a 1-point density defines a stochastic process, and Borland misidentified a translation invariant 1-point density derived from a nonlinear diffusion equation as a conditional probability density. In the Appendix we derive Fokker-Planck pdes and Chapman-Kolmogorov eqns. for stochastic processes with finite memory

No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1

Duality and perfect probability spaces

Abstract. Given probability spaces (Xi, Ai,Pi),i =1,2,let M(P1,P2)denote the set of all probabilities on the product space with marginals P1 and P2 and let h be a measurable function on (X1 × X2, A1 ⊗A2). Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubinˇstein (1958) for the case of compact metric spaces are concerned with the validity of the duality sup { hdP:P∈M(P1,P2)

Regulatory Dynamics on Random Networks: Asymptotic Periodicity and Modularity

We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks.Comment: 23 pages, 3 figure

Population Dynamics in Spatially Heterogeneous Systems with Drift: the generalized contact process

We investigate the time evolution and stationary states of a stochastic, spatially discrete, population model (contact process) with spatial heterogeneity and imposed drift (wind) in one- and two-dimensions. We consider in particular a situation in which space is divided into two regions: an oasis and a desert (low and high death rates). Carrying out computer simulations we find that the population in the (quasi) stationary state will be zero, localized, or delocalized, depending on the values of the drift and other parameters. The phase diagram is similar to that obtained by Nelson and coworkers from a deterministic, spatially continuous model of a bacterial population undergoing convection in a heterogeneous medium.Comment: 8 papes, 12 figure

Finite Size Scaling Analysis of Biased Diffusion on Fractals

Diffusion on a T fractal lattice under the influence of topological biasing fields is studied by finite size scaling methods. This allows to avoid proliferation and singularities which would arise in a renormalization group approach on infinite system as a consequence of logarithmic diffusion. Within the scheme, logarithmic diffusion is proved on the basis of an analysis of various temporal scales such as first passage time moments and survival probability characteristic time. This confirms and puts on firmer basis previous renormalization group results. A careful study of the asymptotic occupation probabilities of different kinds of lattice points allows to elucidate the mechanism of trapping into dangling ends, which is responsible of the logarithmic time dependence of average displacement.Comment: 17 pages TeX, 3 Postscript figure