This paper explores the mixing time of the random transposition walk on permutations with one-sided interval restrictions. In particular, we're interested in the notion of cutoff, a phenomenon which occurs when mixing occurs in a window of order smaller than the mixing time. One of the main tools of the paper is the diagonalization obtained by Hanlon; the use of the spectral information is inspired by the famous paper of Diaconis and Shahshahani on the mixing time of the random transposition walk on the entire symmetric group. The diagonalization allows us to prove chi-squared cutoff for a broad class of one-sided restriction matrices. Furthermore, under an extra condition, the walk also undergoes total variation cutoff. Finally, a large collection of examples which undergo chi-squared cutoff but in which total variation mixing occurs substantially earlier and without cutoff is produced. These results resolve a conjecture of Diaconis and Hanlon from Section 5 of Hanlon's paper.
In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff…
We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff for the one-sided transpo…
We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.
We show that the random transposition walk on the symmetric group $S_n$ has cutoff in separation distance at $\frac{1}{2}n \log n$, by constructing a strong stationary time. The construction involves working with cycle types of permutations and some partition combinatorics.
We prove that the limit profile of star transpositions at time $t= n \log n +cn$ is equal to $d_{\text{T.V.}}(\text{Poiss}(1+e^{-c}), \text{Poiss}(1))$. We prove this by developing a technique for comparing the limit profile behavior of two reversible Markov chains on the same space, that share the same stationary distribution and eigenbasis. We then compare the limit profile of star transpositions to the limit profile of random transpositions, as studied in \cite{Teyssier}, and prove that they have the same limit profile at the respective cutoff times.
We find the total variation mixing time of the interchange process on the dumbbell graph (two complete graphs, $K_n$ and $K_m$, connected by a single edge), and show that this sequence of chains exhibits the cutoff phenomenon precisely when the smaller size $m$ goes to infinity. The mixing time undergoes a phase transition at $m\asymp \sqrt{n}$. We also state a conjecture on when exactly cutoff holds for the interchange process on general graphs. Our proofs use coupling methods, and they also give the mixing time of the simple exclusion process of $k$ labelled particles in the complete graph $…
High-Precision Framework for Expected Hitting Times Analysis in the Dice-Sum Process
Department of Mathematics, Faculty of Science, Mahidol University（玛希隆大学理学院数学系）；Centre of Excellence in Mathematics, MHESI（泰国高等教育与研究创新部数学卓越中心）；Science Division, Mahidol University International College（玛希隆大学国际学院科学学部）
