A generalized @math puzzle' consists of an @math numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' 15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order @math when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any @math with @math , the number of fixed points after @math moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is @math without cut-off. We also prove an upper bound of order @math for the total variation mixing time.
A host of random walks on permutation groups have been studied via a variety of techniques, including the representation theory @cite_8 , @cite_15 , @cite_10 , and couplings @cite_11 . Our method for the upper bound, which uses a three transitive group action, is based on @cite_9 . So far as we are aware, the method of using renewal theory together with a local limit theorem to prove the lower bound, is new. The comparison techniques used in the proof of Theorem are partly based on @cite_17 .
@cite_8 @cite_9 @cite_17 @cite_15 @cite_10 @cite_11
1922672875 2026924428 2963409322 1967591194
We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time @math , the curvature is asymptotically zero for @math and is strictly positive for @math . where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. We study the random walk on the symmetric group (S_n ) generated by the conjugacy class of cycles of length k. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after ( n k n ) steps, uniformly in (k = o(n) ) as (n ). The analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles. We prove a law of large numbers for a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments of some random times depending on the path. It applies in the nonnestling case, but we also provide examples of nestling walks that satisfy our assumptions. The derivation is based on an adaptation, using coupling, of the regeneration argument of Sznitman and Zerner.