Hierarchical self-organisation in complex stochastic systems

Final Report Summary - HISOCS (Hierarchical self-organisation in complex stochastic systems)

Project context and objectives

The research project was concerned with a rigorous mathematical understanding of the emergence of universal hierarchical structures and of the phenomenon of hierarchical self-organisation for two paradigmatic stochastic models of complex systems that involve both short- and long-range interactions: rugged random landscapes with non-hierarchical correlations (spatial complexity), and non-hierarchically interacting diffusions (spatio-temporal complexity).

In the first model, we studied rugged random landscapes that serve as paradigmatic models in physics of complex systems. The general idea is to describe the statics and the dynamics of complex systems, or a subpart of them, by a single point moving in a random potential that encodes the complexity of the system. The concept of rugged landscapes, when combined with ideas and techniques from statistical physics, has led to important breakthroughs in life sciences (fitness landscapes), computer science (hard combinatorial optimisation, machine learning), and economics.

The models of interacting stochastic processes studied in the second phase can be seen as models for the evolution of spatially extended populations. Apart from obvious applications in life sciences (evolutionary biology, genetics, ecology and epidemiology), such and similar models are used in computer science to construct biology-inspired stochastic algorithms with a broad range of applications, e.g. in signal processing and non-linear filtering.

Project work and outcomes

We showed the relevance of hierarchical statistics for a class of non-smooth random landscapes generated by high-dimensional Gaussian fields with isotropic stationary increments on product spaces. We derived a computable saddle-point representation for the free energy of a particle in the landscape, similar to the Parisi formula for the Sherrington-Kirkpatrick (SK) model of a mean-field spin glass. This is an important step towards a rigorous understanding of the geometry of such landscapes. We initiated the analysis of complex-valued landscapes; in particular, we studied fluctuations and phase structure. These landscapes arise naturally via analytic continuation of real-valued landscapes. This seemingly more complicated setting allows for a cleaner identification of analyticity breaking, which is believed to be at the heart of many hierarchical phase-transition phenomena. In such a context, imaginary zeros of the landscape play important role in the identification of phase transitions. A key motivation for the set-up with complex random energies comes from quantum mechanics. There, sums of random exponentials with complex-valued exponents arise naturally in models of interference in inhomogeneous media, and in studies of the quantum Monte Carlo method. We provided a full analysis of the fluctuations of the energy landscape for complex-valued random energy models, which are paradigmatic models of complex-valued random landscapes. We expect that the effects we found are much more universal. We hope that our methods can be extended to a wider class of rugged random landscapes with non-hierarchical correlations.

In the second part, competition between stochastic evolutionary forces - such as resampling, mutation, selection and migration, acting on spatially structured populations - creates intricate space-time patterns of genetic variation. Consider a large system of individuals of multiple types, living in spatially distributed colonies. Let the individuals move randomly between the colonies (= migration). Assume, in addition, that the individuals are subject to stochastic non-local reshuffling-resampling events (= reproduction under a constrained amount of resources). What can we say about the ergodic behaviour of such systems? How does this behaviour depend on the intensities of the competing evolutionary forces? What is the dynamics of the genetic diversity in these systems? Under which circumstances may we expect in the long run local coexistence of individuals of different types within the colonies? Under which circumstances do mono-type clusters of colonies appear? We suggest a class of stochastic models for such systems (based on the so-called Cannings models from population genetics) and obtain a clear-cut criterion for the local coexistence vs. clustering dichotomy. Our analysis relies on renormalisation and duality arguments.

Project results

We achieved the following results:-1) Introduction of a new model for structured populations, based on hierarchically interacting Cannings processes. We substantially increased the scope of previous models by considering interacting systems of jump-processes, i.e. not necessarily diffusions;
-2) Development of duality techniques to study the large space-time scale behaviour of this model;
-3) Derivation of the renormalisation transformation that connects the behaviour on successive space-time scales. In particular, we performed a multi-scale analysis and rigorously derived hierarchical separation of space-time scales;
-4) A full analysis of the iterations of the renormalisation transformation, leading to four universality classes of scaling behaviour as a function of model parameters. The model introduced in the second phase should be seen as an example in a much wider class of models for which similar results may be expected. Work is in progress to explore such extensions.

Final Report Summary - HISOCS (Hierarchical self-organisation in complex stochastic systems)

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