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Register a new userThe Scaling Limit Geometry of Near-Critical 2D Percolation
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We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+lambdadelta^{1/ u}$, with $ u=4/3$, as the lattice spacing $delta to 0$. Our proposed framework extends previous analyses for $p=p_c$, based on $SLE_6$. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of macroscopically pivotal lattice sites and the marked ones are those that actually change state as $lambda$ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.
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Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate $p$, and predator particles spread only to neighboring sites occupied by prey particles at rate $1$, killing the prey particle that existed at that site. It was found that the prey can survive with non-zero probability, if $p>p_c$ with $p_c<1$. Using Monte Carlo simulations on the square lattice, we estimate the value of $p_c = 0.49451 pm 0.00001$, and the critical exponents are consistent with the undirected percolation universality class. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. For all $p < p_c$ in $D$-dimensions, the number of predators in the absorbing configuration has a stretched-exponential distribution in contrast to the exponential distribution in the standard percolation theory. We also study the problem starting from the line initial condition with predator particles on all lattice points of the line $y=0$ and prey particles on the line $y=1$. In this case, for $p_c<p < 1$, the center of mass of the fluctuating prey and predator fronts travel at the same speed. This speed is strictly smaller than the speed of an Eden front with the same value of $p$, but with no predators. At $p=1$, the fronts undergo a depinning transition. The fluctuations of the front follow Kardar-Parisi-Zhang scaling both above and below this depinning transition.
Numerical simulations and finite-size scaling analysis have been carried out to study the percolation behavior of straight rigid rods of length $k$ ($k$-mers) on two-dimensional square lattices. The $k$-mers, containing $k$ identical units (each one occupying a lattice site), were adsorbed at equilibrium on the lattice. The process was monitored by following the probability $R_{L,k}(theta)$ that a lattice composed of $L times L$ sites percolates at a concentration $theta$ of sites occupied by particles of size $k$. A nonmonotonic size dependence was observed for the percolation threshold, which decreases for small particles sizes, goes through a minimum, and finally asymptotically converges towards a definite value for large segments. This striking behavior has been interpreted as a consequence of the isotropic-nematic phase transition occurring in the system for large values of $k$. Finally, the universality class of the model was found to be the same as for the random percolation model.
We reconsider the problem of percolation on an equilibrium random network with degree-degree correlations between nearest-neighboring vertices focusing on critical singularities at a percolation threshold. We obtain criteria for degree-degree correlations to be irrelevant for critical singularities. We present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents.
We investigate the effect of sequentiallydisrupting the shortest path of percolation clusters at criticality by comparing it with the shortest alternative path. We measure the difference in length and the enclosed area between the two paths. The sequential approach allows to study spatial correlations. We find the lengths of the segments of successively constant differences in length to be uncorrelated. Simultaneously, we study the distance between red bonds. We find the probability distributions for the enclosed areas A, the differences in length $Delta l$, and the lengths between the redbonds $l_r$ to follow power law distributions. Using maximum likelihood estimation and extrapolation we find the exponents $beta$ = 1.38 $pm$ 0.03 for $Delta l$, $alpha$ = 1.186 $pm$ 0.008 for A and $delta$ = 1.64 $pm$ 0.025 for thedistribution of $l_r$.
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