We propose an approach to construct Bernoulli trials {X i , i ≥ 1} combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past S i = X 1 + . . . + X i , defines a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials {Y i , i ≥ 1}, called memory switch sequence. We introduce the concept of memory lapses property, which is characterized by intervals of consecutive independent steps in BSRD. The main results include classical limit theorems for a class of linear BSRD. In particular, we obtain a central limit theorem for a class of BSRD which generalizes some previous results in literature. Along the paper, several examples of potential applications are provided.

We show the presence of a first-order phase transition for a ferromagnetic Ising model on Z 2 with a periodical external magnetic field. The external field takes two values h and −h, where h > 0. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides L 1 × L 2 sites, such that the total value of the external field is zero. The phase transition holds if h < 2J L1 + 2J L2 , where J is an interaction constant. We prove the first-order phase transition using the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate.

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We study an urn process containing red and blue balls and two different strategies to reinforce the urn. Namely, a generalized Pólya-type strategy versus an i.i.d. one. At each step, one of the two reinforcement strategies is chosen by flipping a coin. We study the asymptotic behaviour of this urn model, and prove a law of large numbers, a central limit theorem and a functional limit theorem for the proportion of balls into the urn. A phase transition is also stated.

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