# Fast initial conditions for Glauber dynamics

## TL;DR

This paper introduces new methods using information percolation to analyze the mixing times of Glauber dynamics for the 1D Ising model from various initial states, revealing temperature-dependent optimal starting conditions.

## Contribution

It provides the first analysis of mixing times from non-worst-case initial states in Glauber dynamics, showing the alternating initial condition is fastest at high temperatures.

## Key findings

- Alternating initial condition is fastest at high temperatures.
- Mixing time at the optimal initial condition is faster than at infinite temperature.
- The dominant test function varies with temperature, switching from autocorrelation to Hamiltonian.

## Abstract

In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature $\beta$ ranges from 0 to $\beta_0=\frac12\mathrm{arctanh}(\frac13)$. Moreover, the dominant test function depends on the temperature: at $\beta<\beta_0$ it is autocorrelation, whereas at $\beta>\beta_0$ it is the Hamiltonian.