Giovanni M. Cicuta

LARGE N PHASE TRANSITIONS IN LOW DIMENSIONS

Quantum models with fields in matrix representations of classical groups exhibit a high order phase transition in the limit of large order of the group. In zero dimension of space-time the Green functions of the newly found branch are those of a Gaussian model; in one dimension, the physical mass vanishes at the critical point.

Collective synchronization states in arrays of driven colloidal oscillators

The phenomenon of metachronal waves in cilia carpets has been well known for decades; these waves are widespread in biology, and have fundamental physiological importance. While it is accepted that in many cases cilia are mainly coupled together by the hydrodynamic velocity field, a clear understanding of which aspects determine the collective wave properties is lacking. It is a difficult problem, because both the behavior of the individual cilia and their coupling together are nonlinear. In this work, we coarse-grain the degrees of freedom of each cilium into a minimal description in terms of a configuration-based phase oscillator. Driving colloidal particles with optical tweezers, we then experimentally investigate the coupling through hydrodynamics in systems of many oscillators, showing that a collective dynamics emerges. This work generalizes to a wider class of systems our recent finding that the non-equilibrium steady state can be understood based on the equilibrium properties of the system, i.e. the positions and orientations of the active oscillators. In this model system, it is possible to design configurations of oscillators with the desired collective dynamics. The other face of this problem is to relate the collective patterns found in biology to the architecture and behavior of individual active elements.

Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the

Patterns of synchronization in the hydrodynamic coupling of active colloids.

n

Probability density of determinants of random matrices

-th power of a tridiagonal matrix and the enumeration of weighted paths of

Fock space methods and large N

n

Hydrodynamically synchronized states in active colloidal arrays

steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .We present some old and new results in the enumeration of random walks in one dimension, mostly developed in work on enumerative combinatorics. The relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration of paths. It also seems promising for the theory of tridiagonal random matrices.

Multicritical points in matrix models

A system of active colloidal particles driven by harmonic potentials to oscillate about the vertices of a regular polygon, with hydrodynamic coupling between all particles, is described by a piecewise linear model which exhibits various patterns of synchronization. Analytical solutions are obtained for this class of dynamical systems. Depending only on the number of particles, the synchronization occurs into states in which nearest neighbors oscillate in in-phase, antiphase, or phase-locked (time-shifted) trajectories.

Yang-Mills integrals

In this brief note the probability density of a random real, complex and quaternion determinant is rederived using the singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices.In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using the singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices.

A matrix model for random surfaces with dynamical holes

Ideas and techniques (asymptotic decoupling of single-trace subspace, asymptotic operator algebras, duality and role of supersymmetry) relevant in current Fock space investigations of quantum field theories have very simple roles in a class of toy models.