## LMS PiNE Lectures

#### 10, 11, 14 September 2020

A series of 8 live-streamed lectures on

#### Semimartingale methods for Markov chains, interacting particle systems and random growth models

Speakers: Conrado da Costa (Durham), Frankie Higgs (Lancaster), George Liddle (Lancaster), and Chak Hei Lo (Edinburgh).

##### Aims

Sponsored by the LMS and organized by the PiNE research network, this short lecture series is aimed primarily at postgraduate students and early-career researchers in probability theory or neighbouring fields, but all are welcome.

The lectures are in three blocks, with several linking themes running through, but each block should also be essentially self-contained and can be viewed in isolation.

##### Course outline

Foster–Lyapunov methods for Markov chains Chak Hei Lo (3 lectures on 10–11 September).
We will start the course by presenting various results using the semimartingale approach for Markov chains. These results include Foster–Lyapunov criteria by which a suitable Lyapunov function can determine whether a process is transient or recurrent. We will then move on to some applications on these methods, including to some random walks on strips and some interacting particles systems, such as voter models.

Interacting particle systems and martingales Conrado da Costa (3 lectures on 10–11 September).
The purpose of these lectures is to explore martingale methods for the characterization, construction, and analysis of interacting particle systems, including proofs of scaling limit behaviour. The first lecture will focus on the construction of a class of interacting particle systems: a family of coupled birth and death chains. The second lecture will study martingale problems for this model. The third lecture will explore martingale methods for the study of a mean-field voter model.

Planar random growth and scaling limits George Liddle and Frankie Higgs (2 lectures on 14 September).
Conformal growth models are motivated by some real-world growth processes, and are constructed using conformal maps. We will introduce the one-parameter Hastings–Levitov model, which is used to describe Laplacian growth and allows us to vary between off-lattice versions of many well studied models. Then we investigate the "small particle" scaling limit, which often entails finding a martingale and relating its behaviour to its analogue for the proposed continuum limit.

##### How to watch the talks

The talks were delivered live via Zoom and streamed on YouTube.

At the moment the live stream videos are still available via the PiNE YouTube channel.

The pdf slides from the lectures are also available, linked from the schedule below.

## Programme

Note: all times are BST (UTC +1).

#### Thursday 10 September

Chak Hei Lo (University of Edinburgh)
We will start with the definitions of martingales, positive recurrence, null recurrence and transience for countable Markov chains. We introduce the semimartingale approach for studying irreducible Markov chains, and give Foster–Lyapunov criteria for recurrence, transience and positive recurrence. We discuss the main steps in the proofs.
Chak Hei Lo (University of Edinburgh)
Following on from the first lecture, we will look at some simple examples to illustrate the use of Foster–Lyapunov criteria. We will first look at the recurrence classification for $d$-dimensional simple random walk, where our approach provides an alternative proof to the classical Pólya's Theorem. We will then discuss random walks on strips, where the state space is a finite collection of lines or half-lines, and look at the classification in some simple cases.
The purpose of this lecture is to examine the construction of continuous-time Markov processes (CTMP) on discrete state spaces using exponential clocks. We aim to explore the distinction of this set up in comparison to CTMP on finite state spaces. Once we have a basic construction for such processes, we shall move on to consider a couple of tools that allows us to make sense of the evolution of these processes. Those tools are martingales, generators, and semigroups. Towards the end of this lecture we seek to explain how these objects allow us to derive scaling limits of a family of such Markov processes in the framework of the theory of martingale problems due to Stroock and Varadhan.

#### Friday 11 September

Chak Hei Lo (University of Edinburgh)
We will move onto some examples in interacting particles systems. We will introduce the voter model on the integer lattice and discuss its recurrence classification.
The aim of the final two lectures is to introduce some methods commonly used to study a class of random growth processes, in which at each step particles are attached to the boundary of a cluster. In the first lecture we will start by providing examples of the real-world processes that we would like to understand, before describing in detail how models are built in order to study these processes. We will define the Hastings–Levitov model, which is a one-parameter family of models formed using conformal maps that is used to describe Laplacian growth. Finally, we will introduce the different approaches commonly used to find a scaling limit for these processes and consider the phase transitions in the scaling limit of the Hastings–Levitov HL($\alpha$) model as the parameter $\alpha$ varies.