arXiv subjects

Seungjae Son

Publications and source records attributed to Seungjae Son.

5 recordsLinked to original sources

arXiv2026

Convergence of Langevin AIS for multimodal distributions

We study convergence rates of the annealed importance sampling algorithm (Neal '01) combined with Langevin Monte Carlo when the target is a multimodal Gibbs measure. The main result shows that for a fixed error threshold, the time complexity is quadratic in the inverse temperature. We identify a simple and useful quantity that controls the sampling error for AIS in a general setting, and then bound this quantity in our setting using spectral estimates. We also study an autonormalized version and obtain bounds for the time complexity in terms of the inverse temperature.

math.PR↗

arXiv2026

Rapid convergence of tempering chains to multimodal Gibbs measures

We study the spectral gaps of parallel and simulated tempering chains targeting multimodal Gibbs measures. In particular, we consider chains constructed from Metropolis random walks that preserve the Gibbs distributions at a sequence of harmonically spaced temperatures. We prove that their spectral gaps admit polynomial lower bounds of order $11$ and $12$ in terms of the low target temperature. The analysis applies to a broad class of potentials, beyond mixture models, without requiring explicit structural information on the energy landscape. The main idea is to decompose the state space and construct a Lyapunov function based on a suitably perturbed potential, which allows us to establish lower bounds on the local spectral gaps.

math.PR↗

arXiv2025

Quantitative dependence of the Pierrehumbert flow's mixing rate on the amplitude

We quantitatively study the mixing rate of randomly shifted alternating shears on the torus. This flow was introduced by Pierrehumbert '94, and was recently shown to be exponentially mixing. In this work, we quantify the dependence of the exponential mixing rate on the flow amplitude. Our approach is based on constructing an explicit Lyapunov function and a coupling trajectory for the associated two-point Markov chain, together with an application of the quantitative Harris theorem.

math.DS↗

arXiv2025

Exponentially mixing flows with slow enhanced dissipation

Consider a passive scalar which is advected by an incompressible flow $u$ and has small molecular diffusivity $κ$. Previous results show that if $u$ is exponentially mixing and $C^1$, then the dissipation time is $O(|\log κ|^2)$. We produce a family of incompressible flows which are $C^0$ and exponentially mixing, uniformly in $κ$; however have a dissipation time of order $1/κ$ (i.e. exhibits no enhanced dissipation). We also estimate the dissipation time of mixing flows, and obtain improved bounds in terms of the mixing rate with explicit constants, and allow for a time inhomogeneous mixing rate which is typical for random constructions of mixing flows.

math.PR↗

arXiv2025

A Harris theorem for enhanced dissipation, and an example of Pierrehumbert

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If $κ$ denotes the strength of the diffusion, then we show that with probability at least $1 - o(κ^N)$ enhanced dissipation occurs on time scales of order $|\ln κ|$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94.

math.DS↗