A fundamental aim of statistical physics is to predict macroscopic physical phenomena using probabilistic models of microscopic particle–particle interactions. This leads to the need to approximate and sample complicated probability distributions that are often intractable. As such, sampling algorithms have been developed and widely used to enable inference, prediction and model comparison in many different settings.
This workshop aims to give an overview of this topic as well as illustrating scenarios where these techniques are used in practice.
Timetable:
1:30-2:20pm Prof. Mark Jerrum - New Approaches to Perfect Simulation
2:30-3:20pm Dr Michael Faulkner - Phase transitions, metastability and jammed dynamics in statistical physics
3:30-4pm BREAK
4pm-4.50pm Dr Ellen Powell - Geometry of the Gaussian Free Field
5pm Wrap up
5.30pm Close
A fundamental aim of statistical physics is to predict macroscopic physical phenomena using probabilistic models of microscopic particle–particle interactions. This leads to the need to approximate and sample complicated probability distributions that are often intractable. As such, sampling algorithms have been developed and widely used to enable inference, prediction and model comparison in many different settings.
This workshop aims to give an overview of this topic as well as illustrating scenarios where these techniques are used in practice.
Timetable:
1:30-2:20pm Prof. Mark Jerrum - New Approaches to Perfect Simulation
2:30-3:20pm Dr Michael Faulkner - Phase transitions, metastability and jammed dynamics in statistical physics
3:30-4pm BREAK
4pm-4.50pm Dr Ellen Powell - Geometry of the Gaussian Free Field
5pm Wrap up
5.30pm Close
Abstracts:
Mark Jerrum
Title: New Approaches to Perfect Simulation
Abstract. For those of us above a certain age, the possibility of obtaining perfect samples efficiently from a complex probability distribution entered our consciousness with the invention of ‘coupling from the past’ by Propp and Wilson around the mid- nineties. The study of perfect samplers of course has considerable theoretical appeal. But, in addition, their ‘self clocking’ aspect may have practical advantages. For example, to obtain (imperfect) samples by direct Markov chain simulation we need an a priori analytic bound on the mixing time of the Markov chain, which is often very weak; in contrast, a simulation by coupling from the past halts as soon as a perfect sample has been computed. For monotone Markov chains, coupling may occur much earlier than the analytic mixing time bound might suggest.
As coupling from the past is by now well known, I will concentrate on more recent approaches: ‘partial rejection sampling’, which is inspired by Moser and Tardos’ al- gorithmic proof of the Lovász Local Lemma, and ‘lazy depth-first sampling’, which is strikingly simple, but is able to sample configurations of models from statistical physics in the phase uniqueness region.
Michael Faulkner
Title: Phase transitions, metastability and jammed dynamics in statistical physics
Abstract: Sampling algorithms are commonplace in statistics and machine learning – in particular, in Bayesian computation – and have been used for decades to enable inference, prediction and model comparison in many different settings. They are also widely used in statistical physics, where many popular sampling algorithms first originated [1, 2]. At a high level, the goals within each discipline are the same – to sample from and approximate statistical expectations with respect to some probability distribution – but the motivations, nomenclature and methods of explanation differ significantly. This has led to challenges in communicating between the fields, and indeed the fundamental goals of one field are often misunderstood in the other. In this talk, we elucidate statistical physics for the statistician, emphasising that probability models are studied as functions of thermodynamic hyperparameters such as the temperature. This is particularly useful for characterising phase transitions, ie, boundaries in thermodynamic hyperparameter space between distinct thermodynamic phases.
We then move on to sampling algorithms, with a particular focus on the behaviour of the Metropolis algorithm [1] when simulating the 2D Ising and 2DXY models of magnetism. Metropolis dynamics are metastable in the low-temperature phases of these models, mixing between states of equal probability density on a timescale the diverges with system size (proportional to the dimensionality of parameter space). Moreover, the Metropolis algorithm also suffers from the closely related phenomenon of critical slowing down at phase transitions. These jammed dynamics are characterised by divergent (with system size) integrated autocorrelation times, due to a flattening of the target density that essentially results from the system trying to exist simultaneously in both thermodynamic phases. Indeed, these key aspects of statistical physics have led to innovations in sampling algorithms that inform the Bayesian world. In particular, we present the Swendsen—Wang [3], Wolff [4] and event-chain Monte Carlo [5-7] algorithms. The first two simulate the 2D Ising model and were developed in response to the metastability and critical slowing down of the Metropolis algorithm. They circumvent both phenomena to mix with low autocorrelation and independent of system size. We then show that event-chain Monte Carlo similarly circumvents the low-temperature Metropolis metastability of the 2DXY model [7] and discuss its potential utility in bypassing an hypothesised critical slowing down at the phase transition. This talk is based on a recent review paper on the subject [8].
[1] Metropolis et al., J. Chem. Phys. 21 1087 (1953)
[2] Alder & Wainwright, J. Chem. Phys. 27 1208 (1957)
[3] Swendsen & Wang, Phys. Rev. Lett. 58 86 (1987)
[4] Wolff, Phys. Rev. Lett. 62 361 (1989)
[5] Bernard, Krauth & Wilson, Phys. Rev. E 80 056704 (2009)
[6] Michel, Mayer & Krauth, EPL (Europhys. Lett.) 112 20003 (2015)
[7] Faulkner, arXiv:2209.03699 (2022)
[8] Faulkner & Livingstone, arXiv:2209.03699 (2022)
Ellen Powell
Title: Geometry of the Gaussian Free Field
Abstract: I will discuss the strong spatial Markov property (i.e., the theory of “stopping sets”) for the planar Gaussian free field, which is a natural analogue of Brownian motion when time is replaced by a two-dimensional domain. I will describe how this gives rise to natural branching processes encoded by the geometry of the field.