LPTMS PhD Proposal: Exclusion statistics and lattice random walks

Responsable: OUVRY Stéphane + 33 (0)1 69 15 36 30

Thesis proposal :Recently [1] a formula for the algebraic area enumeration of closed random walks on a square lattice has been obtained from the Kreft coefficients which encode the Schrodinger equation of the quantum Hofstadter model.

The Hofstadter model (a charged particle hopping on a square lattice coupled to a perpendicular magnetic field) has a spectrum which is a rare example of a quantum fractal. It happens to be related to closed random walks on a square lattice via a mapping between the n-th moment of the Hofstadter Hamiltonian and the generating function for the enumeration of close lattice walks making n steps and enclosing a given algebraic area. More recently [2] the algebraic area enumeration was generalized to a wider class of random walks and lattices by recognizing the underlying role of exclusion statistics in the enumeration. Several key observations both in [1] and [2] happen to be still incompletely understood and not yet seated on solid mathematical grounds. The enumeration itself has a complexity which increases exponentially with n making it difficult to be used for walks with a large number of steps. The thesis will focus on a better understanding and improving of [1] and [2], in particular simplifying the formula to make it more tractable for large n. Also the investigation of various lattices and random walks will be pushed forward.

[1] S. Ouvry and S. Wu, «The algebraic area of closed lattice random walks » arXiv:1810.04098
[2] S. Ouvry and A. Polychronakos, «Exclusion statistics and lattice random walks » arXiv:1908.00990