Statistical Physics and Interdisciplinary Applications

When we try to study the organization and the properties of ecological systems, non-equilibrium statistical physics is a natural candidate to develop a unified framework for understanding the emergent properties of these kind of systems. Simple interacting particle systems, such as the Voter Model (VM), have found a surprisingly good agreement with empirical data and proved to be a useful null-model that can be treated analytically. Despite the recent progress in this field, still a major issue in ecology and conservation ecology is to understand the effects of habitat fragmentation and heterogeneities on the biodiversity of an ecosystem. Motivated by this open problem, we study the effects of quenched spatial disorder on the long-time behavior of the VM and its nonlinear generalizations.

In my report I will show that the secondary structures of random
heteropolymers undergo a topological transition. Namely, for c less than the critical value the fraction of "active'' nucleotides (which form the base pairs) tends to 1 as the length of the chain goes to infinity, signaling the formation of a virtually "perfect'' secondary structure without gaps. In turn,
for c > c_{cr} always a non-perfect structure with gaps is formed. It was proved mathematically that 2 < c_{cr} < 3; our current research deals with determination a value of transition point; it directly connects with developing new methods to generate random sequences with "effectively" non integer alphabet.
Our previous results have shown that transition point is very close to alphabet used by Nature in real RNAs. Such a critical behavior can point to some statistical exclusivity of natural alphabet.

Spin glasses are ubiquitous models because of their statistical physics properties, and also because they are one of the simplest models where inference algorithms can be tested. We study the application of message passing inference algorithms to finite dimensional Edwards Anderson model. While the naive belief propagation (Bethe approximation) is quite poor in finite dimensional lattice, and does not converge at low temperatures, the generalized belief propagation algorithm shall be a more powerful approximation, but also suffer from convergence problems. We develop a (generalized) message passing algorithm for solving the stationary points of the free energy Edwards Anderson model in 2D and 3D. The messages flow from square plaquettes to links in the graph, and are equivalent to a naive Belief Propagation algorithm in a dual lattice. With the drawback that only the ferromagnetic phase can be explored with our algorithm, it reproduces the results obtained by more general algorithms (GBP-DoubleLoop) in a time that is 3 orders of magnitude smaller. We show the results for the energy at different temperatures for the 2D and 3D models, comparing it with the exact (Monte Carlo) results. The correspondence between exact correlations and inferred correlations of neighboring spins is quite good, specially in 2D model. From a correlation information, a decimation
procedure is implemented to find an approximation to the ground state configuration.

We solve the Edwards-Anderson model (EA) in different Husimi lattices using the cavity method at replica symmetric (RS) and 1-step of replica symmetry breaking (1RSB) levels. We show that, at T = 0, the structure of the solution space depends on the parity of the loop sizes. Husimi lattices with odd loop sizes may have a trivial paramagnetic solution thermodynamically relevant for highly frustrated systems while, in Husimi lattices with even loop sizes, this solution is absent. The range of stability under 1RSB perturbations of this and other RS solutions is computed analytically (when possible) or numerically. We also study the transition from 1RSB solutions to paramagnetic and ferromagnetic RS solutions. Finally we compare the solutions of the EA model in Husimi lattices with that on the (short loops free) Bethe lattices, showing that already for loop sizes of order 8 both models behave similarly.

One of the crucial tasks in many inference problems is the extraction of an underlying sparse graphical model from a given number of high-dimensional measurements. In machine learning, this is frequently achieved using, as a penalty term, the Lp norm of the model parameters, with p≤1 for efficient dilution.
Here we propose a statistical mechanics analysis of the problem in the setting of perceptron memorization and generalization. Using a replica approach, we are able to evaluate the relative performance of naive dilution (obtained by learning without dilution, following by applying a threshold to the model parameters),
L1 dilution (which is frequently used in convex optimization) and L0 dilution (which is optimal but computationally hard to implement). Whereas both Lp diluted approaches clearly outperform the naive
approach, we find a small region where L0 works almost perfectly and strongly outperforms the simpler to implement L1 dilution.