On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases

Abstract

Consider the simple random walk on the n-cycle Z(n). For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) have shown that the log-Sobolev constant alpha is of the same order as the spectral gap lambda. However the exact value of alpha is not known for n>4. (For n = 2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that alpha is 1/2. For n = 3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that alpha = 1/2 log 2 < lambda/2 = 0.75. For n = 4, the fact that alpha = 1/2 follows from n = 2 by tensorization.) Based on an idea that goes back to Rothaus (J. Funct. Anal. 39 (1980) 42; 42 (1981) 110), we prove that if ngreater than or equal to4 is even, then the log-Sobolev constant and the spectral gap satisfy alpha = lambda/2. This implies that alpha = 1/2(1 - cos 2pi/n) when n is even and ngreater than or equal to4. (C) 2003 Elsevier Inc. All rights reserved.