By D. Kannan

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33) An important step in proving Theorem 4 was to show that the exact computation of th e diffusion coefficient (19) and the dependence of th e initial configuration (20) are equivalent. Presumably the same techniques may be applied to show that with the generalized definition of N, given by (29), the dependence on the initial configuration (20) is equivalent to the following identity for the limiting variance lim C:(IE(X£)2 _ (IEX£)2) t - oo t t + = J:- uo(r)(1 - uo(r))dr u+ - u- (34) where u+ and u" are the densities to th e right and left of w(a , t) respectively.

THE ASYMMETRIC EXCLUSION MODEL 15 Acknowledgements Some of the results discussed here have been obtained in collaboration with E. Domany, V. Hakim, S. A. Janowsky, J . 1. Lebowitz, D. Mukamel , V. Pasquier, and E. R. Speer. We thank them as well as D. Foster, C. Godreche, C. Kipnis, K. Mallick, G. Schiitz , and H. Spohn for useful discussions. References I. 2. 3. 4. 5. 6. 7. 8. 9. 10 . 1 I. 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20 . 21. Spitzer, F. (1970) . Interaction of Ma rkov processes. Advances in Mathemat ics 5, 246-290.

Schutz, G . and Domany, E . (1993) . Phase transition s in an exactly soluble one-dimensional exclusion process . Journal of Statistical Physi cs 72, 277 -296. , Evans , M. , and Pasquier, V. (1993) . Exact solution of a ID asymmetric exclu sion model using a matrix formulation. Journal of Physics A : Mathematical and Gen eral 26 , 1493-1517. , Janowsky, S . , Leb owitz, J . , and Speer, E. R. (1993) . Microscopic-shock profiles : exact solution of a non-equilibrium system. Europhysics Letters 22, 651-656; Exact solution of the totally asymmetri c simple exclu sion process: shock profiles.