May 29, 1995 Approach to equilibrium in the symmetric simple exclusion process Nicoletta Cancrini 1,* Antonio Galves 2,** 1 Centre de Physique Th´ eorique - Ecole Polytechnique 91128 Palaiseau Cedex, France. E-mail: cancrini@orphee.polytechnique.fr 2 Instituto de Matem´atica e Estat´ ıstica - Universidade de S˜ao Paulo BP 66281, 05389-970 S˜ ao Paulo SP, Brasil E-mail: galves@ime.usp.br Abstract We obtain an upper bound for the rate of convergence to equilib- rium of the d-dimensional simple symmetric exclusion process start- ing either with a periodic configuration or with a stationary mixing probability distribution. More precisely, calling η * t the process, we show that for all t large enough   P {η * t ∈ Y N }- ρ N    ≤ c d N 2  log t √ t  d , where Y N is the cylindrical set in which all the N sites of a fixed box are occupied. Keywords: Symmetric simple exclusion process. Convergence rate. AMS 1991 Classification numbers. 60K35, 60J15. * Supported by a grant from Consiglio Nazionale delle Ricerche. ** Partially supported by CNPq grants numbers 200487/84-1 and 521738/93-8.