#### Gerardo Barrera: Cut-off phenomenon and an small perturbations of dynamical systems

We study the cut-off phenomenon for a family of an stochastic small perturbations of dynamical systems.
We will focus in a semi-flow of a deterministic differential equation which is perturbed by an small perturbations of a Brownian motion. Under weaker hypothesis on the vector field we can prove that the family of the perturbed stochastic differential equations has cut-off phenomenon.

#### Diego de Bernardini: A decoupling inequality for random interlacements

The random interlacements model was introduced by Alain-Sol Sznitman ([1]) and, roughly
speaking, it is basically defined by a Poisson point process in a space of doubly infinite trajec-
tories in Zd, with d ≥ 3. This model has a positive parameter u, which controls the amount
of trajectories inside the configuration that represents the realization of the process, in such
a way that, the higher the value of u the greater the expected number of observed trajec-
tories. When restricted to a finite subset C ⊂ Z way that independent simple random walk trajectories are started at the boundary of C, the points at which these trajectories are started are randomly chosen in the boundary according to the normalized harmonic measure, and the amount of such trajectories follows a Poisson distribution.
In this context, an interesting and largely investigated issue is related to the character-
ization of the dependence relation (through the covariance) between increasing events with
respect to this model, which are supported on disjoint subsets of the space Z
S. Popov and A. Teixeira introduced, in [3], the soft local times technique, which is a
method to obtain an aproximate stochastic domination between the traces of two Markov
chains, and then they have established an alternative construction of the random interlace-
ments process using this technique.
In this work, we try to use the above mentioned construction of random interlacements
through the soft local times technique, and also the definition of a kind of “joint pivotality”
of collections of excursions of trajectories of the interlacement process on the sets of interest,
aiming to attack the covariance problem. In this way, we obtained a bound on the covariance.
This work is part of the author’s PhD thesis, which was concluded in June, 2014, at
University of Campinas - Brazil, under the supervision of Prof. Dr. Serguei Popov.

#### Clara Fittipaldi: Ray-Knight representation for Crump-Mode-Jagers branching process.

We seek to generalize the work of Lambert et al [1] to obtain a
Ray-Knight representation for general subcritic Crump-Mode-Jagers
branching processes, i.e. to show that this kind of processes can be seen
as local time processes. This result allows us to explode the relationship
between CMJ processes and processor-sharing queues and develop new
techniques to study
them.
[1] A. LAMBERT, F. SIMATOS, and B. ZWART, Scaling Limits via Excursion
Theory: Interplay between Crump–Mode–Jagers branching processes and
Processor-Sharing queue , The Annals of Applied Probability 23 (2013), no.
63, 2357–
2381.

#### Susana Frómeta: Scaling Limit for a Truncated alpha-stable Sandpile

We study the scaling limit of a truncated alpha-stable divisible sandpile, in which each site distributes its excess mass among the lattice with a truncated alpha-stable distribution.

#### Sergio Lopez: Large deviations for proportional fairness allocations

We study the large deviations behavior of the stationary measure of bandwidth-sharing networks functioning under the Proportional fair allocation. We prove that Proportional fair and an associated reversible allocation are geometrically ergodic and have the same large deviations characteristics using Lyapunov functions and martingale arguments. These results comfort the intuition that Proportional fairness is 'close' to allocations of service being insensitive to the service time requirement.

#### Ricardo Misturini: Metastability of the ABC model in the zero-temperature limit

We study the particle system known as the ABC model on a ring in a strongly asymmetric regime. Our main result asserts that the particles almost always form three pure domains (one of each species) and that this segregated shape evolves, in a proper time scale, as a Brownian motion on the circle, which may have a drift.