In this work we prove a shape theorem for a growing set of Simple Random Walks (SRWs) on Z d , known as frog model. The dynamics of this process is described as follows: There are active particles, which perform independent SRWs, and sleeping particles, which do not move. When a sleeping particle is hit by an active particle, it becomes active too. At time 0 all particles are sleeping, except for that placed at the origin. We prove that the set of the original positions of all the active particles, rescaled by the elapsed time, converges to some compact convex set. In some specific cases we are able to identify (at least partially) this set.

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We analyse the Tangle -a DAG-valued stochastic process where new vertices get attached to the graph at Poissonian times, and the attachment's locations are chosen by means of random walks on that graph. These new vertices, also thought of as "transactions", are issued by many players (which are the nodes of the network), independently. The main application of this model is that it is used as a base for the IOTA cryptocurrency system [1]. We prove existence of "almost symmetric" Nash equilibria for the system where a part of players tries to optimize their attachment strategies. Then, we also present simulations that show that the "selfish" players will nevertheless cooperate with the network by choosing attachment strategies that are similar to the "recommended" one.1 this refers to the well-known relation between reversible Markov chains and electric networks, see e.g. the classical book [7] 2 we also cite [2,3,17,22] which deal with other approaches to using DAGs as distributed ledgers 3 that is, the underlying graph is essentially Z + (after discarding finite forks)

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.

We prove decoupling inequalities for the Gaussian free field on Z d , d ≥ 3. As an application, we obtain exponential decay (with logarithmic correction for d = 3) of the connectivity function of excursion sets for large values of the threshold.