Thesis title: Anomalous speed correlations in starling flocks: from microscopic models to field theory
From the analysis of experimental data about starling flocks we can find some key properties of these biological systems. First of all, flocks are highly polarized systems, i.e. every individual's velocity, within one flock, deviates very little from the average flock velocity; secondly, every flocks has a stable average speed of flight of about $12 m/s$, which does not depend on the size of the flock or on the number of individuals that compose the flock; finally, the most peculiar property, fluctuations in the vector velocities and speed fluctuations are scale-free correlated. This means that the correlation length of both directional and speed fluctuations scales with the linear size of the system. The coexistence of scale-free correlations and moderate speed fluctuations is an issue of general relevance for collective behaviour, be it of biological or artificial nature, and yet these two traits set conflicting constraints on the mechanism controlling the speed of each agent, as the factors boosting correlations tend to amplify fluctuations, and vice versa.
In this thesis I present and study a new model that is capable of reproducing all this features without a strong fine-tuning of its parameters. This new theory, the marginal model, relies on a zero temperature critical point that ensures scale-free correlations of all the velocities' degrees of freedom, in the symmetry-broken phase. This property is achieved by confining the speed of each particle with a flat-minimum potential, which means that the minimum of the speed-bounding potential has a vanishing second derivative.
First of all I study the marginal theory at equilibrium, using the mean-field fully-connected approximation and I find that it displays a divergent susceptibility for vanishing temperature, which confirms the idea of a zero temperature critical point. After defining and studying the model from a theoretical point of view, I validate the theory through numerical simulations of the microscopic self-propelled marginal model and by comparison with the experiments. I show that the marginal theory is the only one, so far, compatible with experimental data of natural flocks. Finally, I derive a statistical field theory for the marginal model and I study it using momentum-shell renormalization group (RG) at one loop, for vanishing temperature. I compute the critical exponents that are checked by finite-size scaling analysis of equilibrium on-lattice simulations.