Research Interests

I'm interested in probability theory, statistical mechanics, and related algorithmic questions in theoretical computer science. Some specific topics I have worked on include: quantum spin systems; efficient sampling algorithms; couplings; cluster expansion methods.

Publications and Preprints

• "A New Loop Algorithm with Theoretical Implications" —pending publication (video) 
  • We consider quantum spins with \(S\geq1\), and two-body interactions with \(O(2S+1)\) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for \(S\) sufficiently large. We also prove the existence of a gap for excitations.
• "Critical Parameters for Loop and Bernoulli Percolation" (DOI) (arXiv) 
  • We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter \(\beta\geq0\). Intuitively, larger \(\beta\) means more randomness. In particular, at \(\beta=0\) we start with loops of length 1 and as \(\beta\) crosses a critical value \(\beta_c\), infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with \(p=1-e^{-\beta}\) on the same graph to obtain a lower bound on \(\beta_c\). For those graphs of diverging vertex degree where \(\beta_c\) and the critical parameter for percolation have been calculated explicitly, that inequality has been found to be an equality. In contrast, we show in this paper that for graphs of bounded degree the inequality is strict, i.e. we show existence of an interval of values of \(\beta\) where there are no infinite loops, but infinite percolation clusters almost surely.
• "Dimerization in Quantum Spin Chains with O(n) Symmetry" with J.E. Björnberg, B. Nachtergaele, D. Ueltschi (DOI) (arXiv) (video) 
  • We consider quantum spins with \(S\geq1\), and two-body interactions with \(O(2S+1)\) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for \(S\) sufficiently large. We also prove the existence of a gap for excitations.
• "Bounds on the norm of Wigner-type random matrices" with L. Erdős (DOI) (arXiv) 
  • We consider a Wigner-type ensemble, i.e. large hermitian \(N\times\) random matrices \(H=H^*\) with centered independent entries and with a general matrix of variances \(S_{xy}=\mathbb E|H_{xy}|^2\). The norm of \(H\) is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of \(S\) that substantially improves the earlier bound \(2\|S\|_\infty^{1/2}\) given in [arXiv:1506.05098]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.