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

Consider the simple random walk on the n-cycle Zn. For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) have shown that the log-Sobolev constant α is of the same order as the spectral gap λ. However the exact value of α is not known for n>4. (For n=2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that α is 12. For n=3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that α=12 log 2<λ2=0.75. For n=4, the fact that α=12 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 n⩾4 is even, then the log-Sobolev constant and the spectral gap satisfy α=λ2. This implies that α=12(1−cos 2πn) when n is even and n⩾4.