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Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of "stopping set", not requiring any topological property for the underlying measurable space, as well as on the new concept of a "continuous‐time decision tree", for which we establish several fundamental properties. We apply our findings to the k$$ k $$‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite (2+α)$$ \left(2+\alpha \right) $$‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is 1/2$$ 1/2 $$ for the planar confetti percolation model with bounded, π/2$$ \pi /2 $$‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases. [ABSTRACT FROM AUTHOR]

We present an improvement in estimating the epidemic threshold for a modified susceptible-infected-removed model. We applied the pair heterogeneous mean-field theory on heterogeneous quenched networks. Our theoretical results are compared with extensive Monte Carlo simulations concerning the model dynamics on random regular and power-law networks. The epidemic thresholds obtained using pair heterogeneous mean-field theory agree with simulation results. We analyze the critical scaling of the model as well. At last, epidemic outbreaks emerge for infective probabilities above the critical threshold. [ABSTRACT FROM AUTHOR]

We consider dynamical percolation on the d-dimensional discrete torus Z n d of side length n, where each edge refreshes its status at rate μ = μ n ≤ 1 / 2 to be open with probability p. We study random walk on the torus, where the walker moves at rate 1 / (2d) along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case p < p c (Z d) , the (annealed) mixing time of the walk is Θ (n 2 / μ) , and it was conjectured that in the supercritical case p > p c (Z d) , the mixing time is Θ (n 2 + 1 / μ) ; here the implied constants depend only on d and p. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption θ (p) > 1 / 2 . When θ (p) > 0 , we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are based on percolation results (e.g., the Grimmett–Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time. [ABSTRACT FROM AUTHOR]

We prove that near-critical percolation and dynamical percolation on the triangular lattice ηT have a scaling limit as the mesh = tends to 0, in the "quad-crossing" spaceH of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by "perturbing" the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established. [ABSTRACT FROM AUTHOR]

We present a finite-size scaling theory of the asynchronous susceptible–infected–removed model on scale-free networks, which models epidemic outbreaks and gives a non-vanishing critical threshold. The susceptible–infected–removed model can be mapped in a bond percolation process, as stressed by P. Grassberger, allowing us to compare the critical behavior of site and bond universality classes on networks. We employ a droplet heterogeneous mean-field theory, adding the effect of an external field defined as the initial number of infected individuals. One can choose the external field scaling as N − 1 , where N is the number of network nodes, and compare theoretical results with simulations on the uncorrelated model and Barabasi–Albert networks. The system presents a percolating phase transition where the critical behavior obeys the mean-field universality class, as we show theoretically and by extensive simulations. • We considered the asynchronous SIR model in uncorrelated scale-free networks. • The system presents a percolative phase transition with mean-field exponents. • We show a heterogeneous mean-field theory of the system's finite-size behavior. [ABSTRACT FROM AUTHOR]

Nuclear fuel material is an example of a sintered, porous ceramic material. We formulate a two-dimensional model which couples three physical mechanisms in the material: (scalar) damage accumulation by thermal creep and radiation effects, porosity changes due to the damage, and the time-dependent diffusion of (radiation-induced) gases in the pore system thus created. The most important effect in the dynamics arises from the process where the pore system is swept through the percolation transition. The main conclusions that can be drawn concern the fractional gas release and its dependence on the three effects present in the damage dynamics: creep, radiation-induced bubble formation, and recovery due to bubble closure. In the main, the model reproduces the experimentally observed quick gas release phenomenon qualitatively. [ABSTRACT FROM AUTHOR]

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph $$G$$ are either open or closed and refresh their status at rate $$\mu $$ while at the same time a random walker moves on $$G$$ at rate 1 but only along edges which are open. On the $$d$$ -dimensional torus with side length $$n$$ , we prove that in the subcritical regime, the mixing times for both the full system and the random walker are $$n^2/\mu $$ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice $${\mathbb {Z}}^d$$ holds for this model as well. [ABSTRACT FROM AUTHOR]

Häggström et al. (Ann Inst H Poincaré Probab Stat 33(4):497–528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141–165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time $${t\in D}$$ , in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation. [ABSTRACT FROM AUTHOR]