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Vacancy localisation in the square dimer model, statistics of geodesic in large quadrangulations

Monday 21st April 2008 - 15:30 to 16:30
INI Seminar Room 1

I shall present two independent projects carried out last year, illustrating the interests of a typical combinatorial statistical mechanist.

- I first consider the classical dimer model on a square lattice with a single vacancy, and address the question of the possible motion of the vacancy induced by dimer slidings. Adapting a variety of techniques (Temperley bijection, matrix tree theorem, finite-size analysis, numerical simulations), we find that the vacancy remains localized albeit in a very weak fashion, leading to non-trivial diffusion exponents. [joint work with M. Bowick, E. Guitter and M. Jeng]

- I next consider the statistical properties of geodesics in large random planar quadrangulations. Introducing "spine trees" extending Schaeffer's well-labeled tree construction, we obtain in particular the generating function for quadrangulations with a marked geodesic. We deduce exact statistics for large quadrangulations both in the "local" and "scaling" limits. [joint work with E. Guitter]

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons