Let be a uniformly distributed random -SAT formula with variables and clauses. We prove that the \tt Walksat algorithm from Papadimitriou [On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 1991, pp. 163--169] and Schöning [A probabilistic algorithm for -SAT and constraint satisfaction problems, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 1999, pp. 410--414] finds a satisfying assignment of in polynomial time with high probability if for a certain constant . This is an improvement by a factor of over the best previous analysis of \tt Walksat from Coja-Oghlan et al. [On smoothed -CNF formulas and the Walksat algorithm, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), ACM, New York, SIAM, Philadelphia, 2009, pp. 451--460].