In this article we study the sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. Fo
r both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different dependent percolation models and we discuss some of the problems that can be tackled in a similar fashion. In the last section of the paper we present a long list of open problems that would require new ideas to be attacked.
A hybrid Potts model where a random concentration $p$ of the spins assume $q_0$ states and a random concentration $1-p$ of the spins assume $q>q_0$ states is introduced. It is known that when the system is homogeneous, with an integer spin number $q_
0$ or $q$, it undergoes a second or a first order transition, respectively. It is argued that there is a concentration $p^ast$ such that the transition nature of the model is changed at $p^ast$. This idea is demonstrated analytically and by simulations for two different types of interaction: the usual square lattice nearest neighboring and the mean field all-to-all interaction. Exact expressions for the second order critical line in concentration-temperature parameter space of the mean field model together with some other related critical properties, are derived.
We study a model of opinion exchange in social networks where a state of the world is realized and every agent receives a zero-mean noisy signal of the realized state. It is known from Golub and Jackson that under the DeGroot dynamics agents reach a
consensus which is close to the state of the world when the network is large. The DeGroot dynamics, however, is highly non-robust and the presence of a single `bot that does not adhere to the updating rule, can sway the public consensus to any other value. We introduce a variant of the DeGroot dynamics which we call emph{ $varepsilon$-DeGroot}. The $varepsilon$-DeGroot dynamics approximates the standard DeGroot dynamics and like the DeGroot dynamics it is Markovian and stationary. We show that in contrast to the standard DeGroot dynamics, the $varepsilon$-DeGroot dynamics is highly robust both to the presence of bots and to certain types of misspecifications.
We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy ${f(n)}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that $Cn^{d-2}$ firefig
hters are enough, answering and strengthening a conjecture of Develin and Hartke. We also prove that intermediate growth Cayley graphs do not satisfy polynomial containment, and give explicit lower bounds depending on the growth rate of the group. These bounds can be further improved when more geometric information is available, such as for Grigorchuks group.
We consider the median dynamics process in general graphs. In this model, each vertex has an independent initial opinion uniformly distributed in the interval [0,1] and, with rate one, updates its opinion to coincide with the median of its neighbors.
This process provides a continuous analog of majority dynamics. We deduce properties of median dynamics through this connection and raise new conjectures regarding the behavior of majority dynamics on general graphs. We also prove these conjectures on some graphs where majority dynamics has a simple description.
In [AAV] Amir, Angel and Valk{o} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed
of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-ty
We consider two-dimensional dependent dynamical site percolation where sites perform majority dynamics. We introduce the critical percolation function at time t as the infimum density with which one needs to begin in order to obtain an infinite open
component at time t. We prove that, for any fixed time t, there is no percolation at criticality and that the critical percolation function is continuous. We also prove that, for any positive time, the percolation threshold is strictly smaller than the critical probability for independent site percolation.
We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, out results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive group of autom
orphisms for which the directed balls grow exponentially.
In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order.
Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1,1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of Z, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.
