The following is a list of projects in mathematics I worked on during my PhD.

1. Mixing time of a non-reversible Markov chain on a hypercube.

   In this project, we present a non-reversible Markov chain on the $n$-dimensional hypercube $\{0,1\}{}^n$ which converges in $\mathrm{\Theta }\left(n\right)$ steps. Previously known Markov chains, which are reversible, converge in $\mathrm{\Theta }(n\hspace{0.17em}\log \hspace{0.17em}n)$ steps. This Markov chain alternates between random and deterministic moves, and we prove that the chain also has a cutoff. The deterministic moves correspond to a linear shift register. Here is the publication:

   Fast mixing of a randomized shift-register Markov chain. Journal of Applied Probability. 2023. [pdf]

2. Mixing time of a non-reversible Markov chain on a random graph.

   We present a non-reversible Markov chain on the cycle ${\mathbb{Z}}_n$ with additional random edges and show that it converges in $\mathrm{\Theta }\left(\log \hspace{0.17em}n\right)$ steps with high probability. The random walk is shown to locally be approximated by an auxiliary process, and when this auxiliary process reaches an appropriate entropic threshold, the original Markov chain converges. Most of the previous examples are for reversible chains. One goal of this paper is to show that this idea works also for a non-reversible chain. The other goal is to provide another context where the convergence behavior of a Markov chain can be accelerated by the addition of additional (random) allowable transitions.

3. Mixing time of a Hamiltonian Monte-Carlo Markov chain for the Ising model.

   We present a Hamiltonian Monte-Carlo Markov chain to sample from the Ising model on a complete graph of $n$ vertices, and show that it converges in $O\left(\left(n-\delta \right)\hspace{0.17em}\log \hspace{0.17em}n\right)$ steps at high temperature. Hamiltonian Monte-Carlo Markov chains are most commonly applied for continuous state spaces and have only recently been rigorously analyzed. The main goal of this paper is to apply this Markov chain in a continuous space for sampling from a well-studied discrete probability distribution, namely the Ising model, using a projection. It is worth emphasizing that the projection of the Markov chain onto the discrete space is not a Markov chain.