Absorbing-state phase transition in biased activated random walk

We consider the activated random walk (ARW) model on $\mathbb{Z}^d$, which undergoes a transition from an absorbing regime to a regime of sustained activity. In any dimension we prove that the system is in the active regime when the particle density is less than one, provided that the jump distribution is biased and that the sleeping rate is small enough. This answers a question from Rolla and Sidoravicius (2012) and Dickman, Rolla and Sidoravicius (2010) in the case of biased jump distribution. Furthermore, we prove that the critical density depends on the jump distribution.Comment: In version 5, the upper bound for the critical density in high dimensions has been refined and it has been proved that the critical density depends on the jump distribution. Moreover, some steps of the proof have been simplified. in Electronic Journal of Probability (2016

Critical probabilities and convergence time of Percolation Probabilistic Cellular Automata

This paper considers a class of probabilistic cellular automata undergoing a phase transition with an absorbing state. Denoting by ${\mathcal{U}}(x)$ the neighbourhood of site $x$, the transition probability is $T(\eta_x = 1 | \eta_{{\mathcal{U}}(x)}) = 0$ if $\eta_{{\mathcal{U}}(x)}= \mathbf{0}$ or $p$ otherwise, $\forall x \in \mathbb{Z}$. For any $\mathcal{U}$ there exists a non-trivial critical probability $p_c({\mathcal{U}})$ that separates a phase with an absorbing state from a fluctuating phase. This paper studies how the neighbourhood affects the value of $p_c({\mathcal{U}})$ and provides lower bounds for $p_c({\mathcal{U}})$. Furthermore, by using dynamic renormalization techniques, we prove that the expected convergence time of the processes on a finite space with periodic boundaries grows exponentially (resp. logarithmically) with the system size if $p > p_c$ (resp. $p<p_c$). This provides a partial answer to an open problem in Toom et al. (1990, 1994).Comment: 50 pages, 19 Figure

Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity

We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of `increasing' events, extending previous results of Rolla and Sidoravicius (2012). Our proof method is of independent interest and can be viewed as a reformulation of the `essential enhancements' technique -- which was introduced for percolation -- in the framework of Abelian networks

Uniformly positive correlations in the dimer model and phase transition in lattice permutations in mathbbZd,d>2mathbbZ^d, d > 2, via reflection positivity

Our first main result is that correlations between monomers in the dimer model in ℤd do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices

Dynamical correlations in the escape strategy of Influenza A virus

The evolutionary dynamics of human Influenza A virus presents a challenging theoretical problem. An extremely high mutation rate allows the virus to escape, at each epidemic season, the host immune protection elicited by previous infections. At the same time, at each given epidemic season a single quasi-species, that is a set of closely related strains, is observed. A non-trivial relation between the genetic (i.e., at the sequence level) and the antigenic (i.e., related to the host immune response) distances can shed light into this puzzle. In this paper we introduce a model in which, in accordance with experimental observations, a simple interaction rule based on spatial correlations among point mutations dynamically defines an immunity space in the space of sequences. We investigate the static and dynamic structure of this space and we discuss how it affects the dynamics of the virus-host interaction. Interestingly we observe a staggered time structure in the virus evolution as in the real Influenza evolutionary dynamics.Comment: 14 pages, 5 figures; main paper for the supplementary info in arXiv:1303.595

Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths

Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations

We consider nearest neighbour spatial random permutations on Zd. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, α, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of α (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of α in great generality