Understanding the statistical properties of complex systems and of their predominant out-of-equilibrium dynamics is central to a large set of disparate fields, including statistical and high-energy physics, computer science, economy and finance, biology and ecology.
Spin-glasses and glass physics offer a prototypical example of complex systems where slow dynamics is originated by a complicated non-convex structure of the energy landscape. Their investigation led to a plethora of studies carrying multiple solid results about the intimate connection between the structure of the energy landscape and main traits of dynamical phenomenology, which can nowadays be applied to different systems characterized by slow dynamics and rough cost landscapes.
In recent years due to the emergence of new questions and applications, research on this topic has boomed showing the extent of the potentialities of theoretical physics approaches in the corresponding fields.
This project aims at making further steps in this direction in the different fields ranging from the prototypical example of spin glasses to modern applications in ecology, statistical models of learning and artificial neural networks, dynamics in quantum systems.
We believe that our project can change the state of the art in many directions.
Building on results that show similarities between learning and glass dynamis, we plan to develop a more detailed study of the Loss landscape in terms of the local geometrical properties of its stationary points visited during learning. This study will allow the formulation of a coarse grained description of such landscape and its use to model learning dynamics in terms of mesoscopic simple trap/step models where collections of configurations associated to different levels of a cost function can be sequentially visited by a stochastic process whose typical behaviour can be predicted [Cugliandolo1993, Barrat1995, Bouchaud1995].
Following the recent flourishing of models of potential ecological relevance [Biroli2018, Altieri2019] and of statistical mechanics approaches [Roy2019, Mambuca2020] to study them, we plan to make significant progresses in the field first by building on recent results about the predictions for the MacArthur consumer resource model [Altieri2019] and the important development of analytic tools to access the statistical properties of stationary points of the dynamics (Kac-Rice formalism, replica computations) [Bray2007, Auffinger2013, Fyodorov2015]. In particular, we plan to study the properties of equilibria of models where common resources lead to (symmetric) interactions between more than two species to analyse the impact of their contribution and rationalise experimental results [Friedman2017].
Beyond the approximation of symmetric interactions, the study of the rich dynamical phenomenology of models of ecosystems will develop from recent results [Mambuca2020] obtained in the linear approximation and by using random matrix theory that hint towards important differences on the stability properties depending on the type of interactions (antagonistic vs mutualistic and competitive), when species interact with a limited number of other species. Further studies will be carried out to assess whether the qualitative differences expected will appear in the dynamical behaviour of the model ecosystem beyond the linear approximation in the dynamics and the simplifications imposed by the random matrix approach.
A more general approach, going under the name of dynamic mean field theory, to the study of the dynamical behaviour of ecosystems has been recently developed [Roy2019] to describe ecosystems where species interact with all the others. Within this approach, a stochastic process representative of the evolution of any species in the ecosystem is obtained by iteratively adding time auto-correlation induced by the interaction with other species. We plan to work on an extension of its application to the opposite limit where species interact with few others exploiting the absence of short range correlations in tree-like local structure of the resulting interaction graph.
As far as spin glasses are concerned we will orient this project to try and understand theoretically the behavior of different experimental protocols (zero field cooling and thermoremanent magnetization, with different variations).
During this project we will also study realistic statistical models of memory, trying to understand their learning skills.
Also we will set up interactions with experimental groups to further develop our theoretical methods to determine the exponents of the diffusion of non-Brownian travelers.
Recently we have introduced a class of first order quantum phase transitions characterized by a condensation in the space of states [OP2021], something reminiscent of Anderson¿s orthogonality catastrophe [Anderson1967]. The outlined mechanism has been tested on a variety of benchmark lattice models, including spin systems, free fermions with nonuniform fields, short-range interacting fermions and hard-core bosons. We think that the developed analytical tools, actually a powerful existence theorem of the transition with the possibility to exactly bound the transition parameter, will allow us to address also the long standing problem of Wigner crystallization. The quantum phase transition of free electrons into a lattice crystal, suggested by Wigner in 1934, has so far run away from any rigorous proof, yet the recent observation of Wigner crystal in carbon nanotubes aimed at realize solid state quantum computing with long coherence times [DB2008], make this phenomenon very timely.
REFERENCES
Some of the references cited in this section are listed in the previous section.
[OP2021] M. Ostilli, C. Presilla, "First-order quantum phase transitions as condensations in the space of states", J. Phys. A: Math. Theor. 54, 055005 (2021).
[Anderson1967] P.W. Anderson, "Infrared catastrophe in a Fermi gas with local scattering potentials" Phys. Rev. Lett. 18 1049 (1967).
[DB2008] V.V. Deshpande, M. Bockrath, "The one-dimensional Wigner crystal in carbon nanotubes", Nature Physics 4, 314-318 (2008).