In this work, we study the problem of getting quasi-additive bounds for the Hamiltonian of the long range Ising model, when the two-body interaction term decays proportionally to $\frac{1}{d^{2-\alpha }}$ , $\alpha \in \left(\mathrm{0,1}\right)$ . We revisit the paper by Cassandro et al. [J. Math. Phys. 46, 053305 (2005)] where they extend to the case $\alpha \in \left[0,\frac{ln3}{ln2}-1\right)$ the result of the existence of a phase transition by using a Peierls argument given by Fröhlich and Spencer [Commun. Math. Phys. 84, 87–101 (1982)] for $\alpha =0$ . The main arguments of Cassandro et al. [J. Math. Phys. 46, 053305 (2005)] are based in a quasi-additive decomposition of the Hamiltonian in terms of hierarchical structures called triangles and contours, which are related to the original definition of contours introduced by Fröhlich and Spencer [Commun. Math. Phys. 84, 87–101 (1982)]. In this work, we study the existence of a quasi-additive decomposition of the Hamiltonian in terms of the contours defined in the work of Cassandro et al. [J. Math. Phys. 46, 053305 (2005)]. The most relevant result obtained is Theorem 4.3 where we show that there is a quasi-additive decomposition for the Hamiltonian in terms of contours when $\alpha \in \left[\mathrm{0,1}\right)$ but not in terms of triangles. The fact that it cannot be a quasi-additive bound in terms of triangles lead to a very interesting maximization problem whose maximizer is related to a discrete Cantor set. As a consequence of the quasi-additive bounds, we prove that we can generalise the [Cassandro et al., J. Math. Phys. 46, 053305 (2005)] result, that is, a Peierls argument, to the whole interval $\alpha \in \left[\mathrm{0,1}\right)$ . We also state here the result of Cassandro et al. [Commun. Math. Phys. 327, 951–991 (2014)] about cluster expansions which implies that Theorem 2.4 that concerns interfaces and Theorem 2.5 that concerns n point truncated correlation functions in Cassandro et al. [Commun. Math. Phys. 327, 951–991 (2014)] are valid for all $\alpha \in \left[\mathrm{0,1}\right)$ instead of only $\alpha \in \left[0,\frac{ln3}{ln2}-1\right)$ .

Consider a topologically transitive countable Markov shift $\Sigma $ and a summable locally constant potential $\phi $ with finite Gurevich pressure and $\mathrm {Var}_1(\phi ) < \infty $ . We prove the existence of the limit $\lim _{t \to \infty } \mu _t$ in the weak $^\star $ topology, where $\mu _t$ is the unique equilibrium state associated to the potential $t\phi $ . In addition, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.

In this article we provide new results for the asymptotic behavior of a time-fractional birth and death process , whose transition probabilities are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23].

We study the $R_\beta$ -positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(\mu_{\beta}(\!\cdot\!))_{\beta \geq 0}$ at inverse temperature $\beta$ associated to a family of nearest-neighbor matrices $(Q_{\beta})_{\beta \geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $\beta$ , and (ii) the existence of weak limits as $\beta \to \infty$ . Some application examples are revised to put in context the main results of this work.