# Mathematics Colloquia and Seminars

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### Asymmetric diffusions and the spectral gap of XXZ quantum Heisenberg models

**Mathematical Physics & Probability**

Speaker: | Fabio Martinelli, U. Roma III / UC Berkeley |

Location: | 693 Kerr |

Start time: | Tue, Mar 11 2003, 3:10PM |

Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, I will consider an anisotropic simple exclusion process with N particles in a rectangle of the square lattice. Every particle at a given row tries to jump to an arbitrary empty site of the upper row q and to the lower row with rate 1/q where 0<q<1 is a measure of the drift driving the particles towards the bottom of the rectangle. It is shown that the spectral gap of the generator is uniformly positive in N and in the size of the rectangle. The proof is inspired by a recent interesting technique envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for the non linear Boltzmann equation. The result will then be applied to prove precise upper and lower bounds on the energy gap for the half integer spin S XXZ chain and for the 111 interface XXZ Heisenberg model, thus generalizing previous results valid only for spin 1/2. Joint work with P. Caputo.