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Scientific Report 2007-2009
Theoretical physics
of generalized formulations of differential geometry.
Discussing now about our results in the Physics of Disordered and Complex Systems implies
a sizable ideal jump (even if, obviously, complexity is potentially at the root of understanding
of a large number of issues on the scale of the universe). When we discuss these researches we
have to keep in mind strongly the tight relations with the work described in the condensed matter
and biophysics section, since connections are, in this case, sometimes really strong. Our researches
about the “Statistical Mechanics of disordered systems and renormalization group” are described
in [T12]. In the last few years our researchers have performed several numerical studies of the
three-dimensional Edwards-Anderson model, a prototypical finite dimensional spin glass. The use
of the most advanced numerical techniques — the parallel-tempering method, multi-spin coding,
cluster algorithms, etc. — and of very fast computers has allowed to address long-standing problems
and to obtain several new and important results. A significant improvement of the quality
of numerical simulations of random systems has been obtained by developing a new dedicated
machine (JANUS) in collaboration with the University of Ferrara and several Spanish research
groups. JANUS is a modular, massively parallel, and reconfigurable FPGA-based computing system.
We only quote one further result among many: a new one-dimensional spin-glass model with
long-range interactions has been introduced. The interaction between two spins a distance r apart
is either ±1 with a probability that decays with r as 1/r ρ , or zero. Depending on the exponent ρ,
the model may or may not show mean-field behavior: for ρ ≤ 4/3 the mean-field approximation
is exact, for ρ > 2 no phase transition occurs, while in between the behavior is nontrivial.
Spin glass physics has received, with replica symmetry breaking and the understanding of the
spin glass, many state phase, many crucial contributions from researchers of our group, and this
study is progressing at a fast rate.
Our researchers have studied the physics of “The Glassy State” [T13]. A one-dimensional version
of the Derridas Random Energy Model has been analyzed. The Random Energy Model, being a
long range model, has a clear random first order transition. In the 1D model a length (proportional
to the system size, as in the Kac limit) has been introduced, such that interactions are Random
Energy Model-like on smaller scales. They have, as well, dedicated a large effort in recent years on
the study of glasses of hard spheres. A system of monodisperse hard spheres is maybe the simplest
showing most of the glass phenomenology and can be thus considered as a prototypical model:
the interest to study it is very large. Similar ideas have been applied to “optimization problems
and message passing algorithms” in [T14]. Among optimization problems, a quite general class
is formed by Constraint Satisfaction Problems where a set of constraints is given, and where the
constraints must be satisfied by a proper assignment of the variables. Our researchers have been
able to solve this kind of models in the case where the constraints are generated independently,
which actually correspond to defining the model on a random graph. A very important aspect of
the message passing algorithms that our researchers have started to investigate recently is their
use on non-random graphs, that is graphs with many short loops and topological motifs. An
interesting example is given by the problem of ranking graphs nodes, i.e. to uncover which nodes
are the most important in the graph topology (a straightforward application being the ranking of
web pages). A new message passing algorithm that ranks nodes depending on how many loops
pass through that node has been introduced. The last contribution to this research line is about
neural networks: “From Artificial Neural Networks to Neurobiology”, in [T15]. One main topic
investigated by our group is synchronization. On one side neurons can be described by a set
of first order linear differential equation with an interaction matrix with Gaussian distributed
random elements: on the other side a more biological approach has led to the modeling of the
behavior of oxytocin neurons of the hypothalamus when they emit the oxytocin hormone. Also
the problem of the control of the movements of the eye (saccadic movements) has been considered.
Again smoothly, through the last research activity we have discussed, we can shift to the last
of the four subjects we describe here, Mathematical Physics. We will see that also here we
will encounter a large variety of interesting researches, developing a large number of new ideas in
Sapienza Università di Roma 22 Dipartimento di Fisica