Related papers: Mixing of the upper triangular matrix walk

The (standard) average mixing matrix of a continuous-time quantum walk is computed by taking the expected value of the mixing matrices of the walk under the uniform sampling distribution on the real line. In this paper we consider…

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first…

A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit a tractable Laplace transform (probability generating…

Several cycle lexicographical orders are found to describe the relative likelihood of elements of the random walks on the symmetric group generated by the conjugacy classes of transpositions, 3-cycles, and n-cycles. Spectral analysis finds…

Quantum random walk finds application in efficient quantum algorithms as well as in quantum network theory. Here we study the mixing time of a discrete quantum walk over a square lattice in presence percolation and decoherence. We consider…

We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $\mu \ll 1/n$, and is then declared open with probability $p = \lambda/n$ where $\lambda > 1$. We study a random walk on this…

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…

We study an example of a {\em hit-and-run} random walk on the symmetric group $\mathbf S_n$. Our starting point is the well understood {\em top-to-random} shuffle. In the hit-and-run version, at each {\em single step}, after picking the…

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths,…

We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one…

Mixing properties of discrete-time quantum walks on two-dimensional grids with torus-like boundary conditions are analyzed, focusing on their connection to the complexity of the corresponding abstract search algorithm. In particular, an…

We consider random walks on a tree $G=(V,E)$ with stationary distribution $\pi_v = \mathrm{deg}(v)/2|E|$ for $v \in V$. Let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at vertex $v$ to…

In [3] the radius of convergence of the generating function of the collision local time of two independent copies of an irreducible, symmetric and transient random walk on Zd, d \geq 1, was studied. Two versions were considered: z1, the…

Consider a random walk on a tree $G=(V,E)$. For $v,w \in V$, let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at $v$ to reach $w$, and let $\pi_v = \mathrm{deg}(v)/2|E|$ denote the…

The Diaconis--Gangolli random walk is an algorithm that generates an almost uniform random graph with prescribed degrees. In this paper, we study the mixing time of the Diaconis--Gangolli random walk restricted on $n\times n$ contingency…

We analyze a fairly standard idealization of Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G. It is found that, with high probability, a collision occurs in $O(\sqrt{|G|\log |G| \log \log |G|})$ steps, not far…

The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82},…

The transpose top-$2$ with random shuffle (J. Theoret. Probab., 2020) is a lazy random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(\star,n-1,n)$ and $(\star,n,n-1)$. We obtain the limit profile of this random…

Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schr\"{o}der, and Motzkin number sequences…

In this paper we consider an excited random walk on $\mathbb{Z}$ in identically piled periodic environment. This is a discrete time process on $\mathbb{Z}$ defined by parameters $(p_1,\dots p_M) \in [0,1]^M$ for some positive integer $M$,…