Seminars

Justin Forlano (Monash): Invariant Gibbs dynamics for the hyperbolic sinh-Gordon model

5 August

Whilst there has been a wealth of progress on random dispersive equations with polynomial nonlinearities, the non-polynomial case remains much less developed. In this talk, I will discuss recent progress, joint with Y. Zine (EPFL), on the well-posedness (and invariance of the Gibbs measure) for the two-dimensional singular stochastic periodic damped Klein-Gordon equation with exponential-type nonlinearities; a setting which covers the so-called hyperbolic Liouville and hyperbolic sinh-Gordon models. We develop a novel physical space framework for wave equations with non-negative and non-polynomial nonlinearities, which goes beyond the traditional L^2-framework, and obtain the first results for the hyperbolic sinh-Gordon model and improved results for the hyperbolic Liouville model.

Emanuel József Godfried (University of Melbourne): Escape of hyperbolic Brownian motion

2 September

We discuss properties of hyperbolic Brownian motion and compare to Euclidean Brownian motion. We show that hyperbolic Brownian motion is not diffusive for long time and discuss possible resolutions to obtain a random process which is diffusive for every time t. We also discuss properties of the Dirichlet eigenvalues of Laplacian on a smooth compact subset of hyperbolic space and show how the eigenvalues converge to the spectrum of the Laplacian on hyperbolic space.

Benoit Sixte Corsini (National University of Singapore): Local limit of Prim's algorithm

9 September

In this talk, I will present a result regarding the local evolution of Prim's algorithm seen as a random process on graphs. To do so, I will start by introducing the concept of local limit for a sequence of graphs, before defining Prim's algorithm and showing its relation to the minimum spanning tree. Finally, I will combine these two concepts to provide a representation of the limit, in particular its relation to invasion percolation and the percolation levels of a graph.

Adair da Silva Neto (University of Melbourne): Fluctuation Asymptotics for the Parabolic Anderson Model on theHeisenberg Group

23 September

The Parabolic Anderson Model is a stochastic heat equation known for its intermittent behaviour and nontrivial long-time fluctuations. While the discrete setting is by now well understood, much less is known in the continuum. In this talk, we investigate the model on the Heisenberg group and analyse the variational problem determining the fluctuation constant. The results reveal how the sub-Riemannian structure and noncommutative geometry influence the asymptotic behaviour of the solution. This is based on joint work with Marco Carfagnini and Xi Geng.

Giuseppe D’Onofrio (Polytechnic University of Turin) : A non-local Jacobi operator and the related first-passage-time problem

12 January

Motivated by some applications in the context of neuronal modeling, we introduce downward jumps in a class of Jacobi stochastic processes. The jumps are state dependent both in frequency and amplitude. The properties of the resulting Levy-type process are investigated and general results for its first-passage time, T, through a constant boundary are developed. We characterize the Laplace transform of T in terms of generalized hypergeometric functions that we introduce, getting a new closed-form expression of the
expectation of T. We follow an original approach based on intertwining relations, which have been recently established, between the semigroup of classical Jacobi processes with the one of their generalized versions. A numerical investigation of these results is also considered for different choices of the involved parameters and of the jumps distributions. Based on joint works with Pierre Patie (Cornell University), Laura Sacerdote (Università di Torino) and Alessandro Lanteri (Università di Torino).

Anna Paola Todino (University of Eastern Piedmont): Geometric Functionals of Random Spherical Harmonics: From Wiener Chaosto a Generalized Framework

12 January

In the last decades, much effort has been devoted to the analysis in the high-frequency behavior of geometric functionals (Lipschitz-Killing curvatures) for the excursion sets of Gaussian random fields. In this talk, we attempt to provide an overview of the statistical properties of these functionals, with particular attention to the nodal volume of random spherical eigenfunctions. We begin by examining the two-dimensional case (the nodal length) and then we extend our analysis to higher dimensions. A key tool in this investigation is the Wiener chaos expansion, which allows us to analyze the fluctuations and limiting distributions of these geometric quantities. However, the application of this technique, in its standard formulation, presents technical obstacles and limitations. After discussing these constraints, in the final part of the talk, we will introduce a new version of the chaos expansion that overcomes some of these difficulties. Crucially, this new framework is not limited to the spherical context or the eigenfunctions of the Laplacian. Instead, it applies to a more general class of Gaussian random fields on a general Riemannian manifold.

Nathan Ross (University of Melbourne): Detecting correlation in uniform attachment trees

18 March

Networks are a fundamental data structure for modelling association between pairs of agents in a system. There has been significant recent interest in the problem of detecting correlation between different networks, with applications in, for example, de-anonymizing social networks, and inferring function in biological networks.

A first fundamental approach to the problem is to derive sharp information-theoretic thresholds for detection in specific models. Most of the literature thus far has focussed on Erdős–Rényi and stochastic block models, where the random edges between pairs of nodes are independent. While these models form a good theoretical test bed, it is well-known that they are not good fits for empirical network data, and so extending this line of research to other network models is an important research avenue.

In this talk, we introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. We discuss the question of how well the correlation can be detected from a single pair of unlabeled correlated UA trees, and show that this can be done with probability tending to one as the size of the trees goes to infinity, by constructing a statistic of the unlabeled trees that converges to the correlation parameter.

The construction of our statistic relies on two key ideas. The first is that we can use a notion of centrality to identify subsets of vertices of each tree whose intersection has a sufficient number of common early vertices. The second idea is that across different scales, it is possible to approximately determine the labels of vertices that have attached to these early vertices, using the sizes of fringe subtrees. Our analysis includes quantitative bounds on the fraction of early vertices that remain most central, which may be of independent interest.

Pierre Perruchaud (Monash): Brownian loop soup and Gaussian free field for quantum field theory

1 April

In the standard model of particle physics, the main mathematical protagonists are sections of vector bundles (modelling e.g. the electron
field) and connections on these bundles (modelling e.g. the electromagnetic potential). Although they are well-understood by the geometry community, theoretical physics describes some sort of evolution equation for those, and the question of existence of a solution is widely open. It is believed that one can approach problems of this type by constructing a random measure on a space of pairs (section, connection). In this talk, we will introduce all these objects, and see that one can construct such measures using seemingly unrelated random objects, namely collections of Brownian loops on the base space.

Marco Carfagnini (University of Melbourne): Berry-Heisenberg random waves

22 April

In the 70s, Berry argued that in the high-energy limit wave functions locally look like random superpositions of independent plane waves,
having all the same wavenumber. He introduced a Gaussian random field whose sample paths are generalized Laplace eigenfunctions. The aim of this talk is to present a similar model for the sub-Laplacian on the Heisenberg group, which is the analogue of the Euclidean space in sub-Riemannian geometry. This is based on a current project with Anna Paola Todino from the University of Eastern Piedmont, Italy, and it combines ideas from functional analysis, representation theory, and stationary random fields.

Frank den Hollander (Leiden University): Challenges in Co-evolution on networks

20 May

In this talk, I present an overview of what is known and is not known about random processes on dynamic random graphs in co-evolution, i.e., with mutual feedback. The literature offers plenty of heuristics, simulations and conjectures, but so far mathematical results are extremely scarce. The focus will be on random graphs in which vertices can have one of two possible opinions. Pairs of vertices connected by an edge share their opinions according to a rate that may depend on the size of the graph. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same opinion or not. I will exhibit joint evolution equations and describe the occurrence of phase transitions between consensus and polarisation.

Jie Yen Fan (Monash): Estimation in age-and-population-dependent models

20 August

We consider general age-and-population-dependent population systems, where individual birth and death rates depend not only on age but also on the overall population composition. These systems can be modelled as measure-valued stochastic processes. Functional Law of Large Numbers and Central Limit Theorem with large carrying capacity can be established. These results, with appropriate test functions, allow us to estimate the demographic rates and obtain their confidence bounds. Joint work with Kais Hamza, Fima Klebaner and Ziwen Zhong.

Stanley Luk (UTS): New Analogies for Rough Paths and Signatures

3 September

Rough path theory is an integral calculus on continuous functions with finite p-variations, through which one is able to integrate with respect to sample paths of various stochastic processes such as Brownian motion, fractional Brownian motions, to name a few. As an alternative to the study ofSDEs, in this deterministic framework one solves ODEs driven by such rough paths, called Rough Differential Equations (RDEs). We present aspects of our work on generalising the algebraic and geometric setting where rough paths and signatures of paths live. Morally, an irregular sample path cannot be uniquely determined by its trace alone and requires additional higher order geometric data living in some rigid algebra structure in order to be usefully defined.Although the main ingredients are Hopf algebras and the differential geometry of fibre bundles, we will eschew the technical details for simple examples of other objects this general setting encompasses. There are links to group theory, quantum algebra, and Lie groups in the geometry of physics, all of which provide a rich source of analogies which can be related back to rough paths. With this, we hope to provide new insights into what a rough path fundamentally is.

Andriy Olenko (La Trobe): On stochastic partial differential equations with random initial conditions

24 September

In this talk, we address two problems about solutions of stochastic partial differential equations with random initial conditions.

The first part focuses on the Cauchy problem for random fields generated by partial differential equations on the unit sphere. The exact solution as a series expansion in terms of spherical harmonics is given. An approximation to the solution is provided and analysed by finitely truncating the series expansion. We establish upper bounds for the convergence rates of approximation errors and investigate the smoothness of both the exact and truncated solutions. In particular, we show how the sample Hölder continuity of the resulting spherical fields is determined by the decay of the angular power spectrum.

The second part examines multiscaling limit theorems for renormalised solutions of higher-order heat equations with initial conditions given by random processes exhibiting cyclic long-range dependence. We derive the spectral and covariance representations of the corresponding limit fields and explain why analogous limit results are not valid in subordinated settings with Hermite rank greater than one. Numeric examples will be presented to illustrate the theoretical findings.

The talk is based on joint results in
[1] Leonenko, N.,  Olenko, A., Vaz, J. (2024) On fractional spherically
restricted hyperbolic diffusion random field. Commun. Nonlinear Sci. Numer.
Simul., 131, 107866
[2] Broadbridge, P., Donhauzer, I. Olenko, A. (2024) Stochastic diffusion
within expanding space–time. Z. Angew. Math. Phys. 75, 42
[3] Alghamdi, M.M., Leonenko, N.,  Olenko, A. (2025) Multiscaling limit
theorems for stochastic FPDE with cyclic long-range dependence.
arXiv:2409.09215

Sheng Wang (University of Melbourne): Topics in Rough Path Theory and Stochastic Analysis

8 October

Firstly, we will briefly present the main results of three papers completed during the last two years of my PhD: Cartan's Path Development, the Logarithmic Signature and a Conjecture of Lyons–Sidorova, Asymptotics of the Solution to the Parabolic Anderson Model in Hyperbolic Space, and Hitting Probability for Rough Differential Equations Driven by Fractional Brownian Motion.

Afterwards, we will outline the methodology and describe our recent progress on the first project. The signature transform, which is defined in terms of iterated path integrals, provides a faithful representation of the group of tree-reduced geometric rough paths. It was conjectured by T. Lyons and N. Sidorova that the only tree-reduced paths with bounded variation (BV) whose logarithmic signature can have infinite radius of convergence (R.O.C.) are straight lines. Our main result is that if the logarithmic signature has infinite R.O.C., the signature coefficients must satisfy an infinite system of rigid algebraic identities defined in terms of iterated integrals along complex exponential one-forms. These iterated integral identities impose strong geometric constraints on the underlying path, and in some special situations, confirm the conjecture. As a non-trivial application of our integral identities, we prove a weak version of the conjecture, which asserts that if the logarithmic signature of a BV path has infinite radius of convergence over all sub-intervals of time, the underlying path must live on a straight line. Our methodology relies on Cartan's path development onto the complex semisimple Lie algebras. The special root patterns allow one to project the infinite-dimensional Baker-Campbell-Hausdorff (BCH) formula in a very special finite dimensional manner to yield meaningful quantitative relations between BCH-type singularities and the vanishing of certain iterated
path integrals.

Sophie Hautphenne (University of Melbourne): Consistent estimation in subcritical birth-and-death processes

19 November

Birth-and-death processes are the simplest continuous-time branching processes and are widely used in applications. Subcritical birth-and-death processes model declining populations that would naturally die out without conservation efforts. When we apply these processes to model endangered populations, the data should be interpreted as generated by the process conditioned on survival, and consistent estimators are desirable.

In this talk, we propose the first consistent estimators for parameters of subcritical birth-and-death processes, based on continuous observation of a single non-extinct trajectory of the process. The idea behind their construction stems from a spine decomposition of subcritical branching processes conditioned to survive in the distant future.

Ben Morris (UC Davis): Mixing time of the torus shuffle

20 November

In 1988, Diaconis introduced the following model of card shuffling. Cards are arranged in an n by n grid. Each step, choose a random row or column and cyclically rotate it by one unit in a random direction. He conjectured that the mixing time is O(n^3 log n). We obtain a bound that is within a poly log factor of the conjecture.

Joint work with Olena Blumberg and Alto Senda.

Gleb Smirnov (ANU): Regularizing random points by moving a few

15 April

A well-known fact in probability is that a set of n independent random points in [0,1] has star discrepancy of order O(1/sqrt(n)). In this talk, we shall see how to improve the discrepancy to O(1/m) by moving at most O(m) points. The proof is constructive and consists of a simple, data-adaptive algorithm with small time complexity. This is a joint work with Roman Vershynin.

Stephen Muirhead (Monash): On the number of clusters in a strongly correlated percolation model

27 May

It has been known since the work of Penrose and Zhang in the 2000s that the number of clusters in classical i.i.d. lattice percolation models, restricted to large domains, has an asymptotically Gaussian distribution for any parameter of the model. Is this also true for percolation models with correlations?

In this talk we consider a percolation model defined by the excursion sets of the Gaussian free field on the lattice Z^d, d \ge 3. In this model the level plays the role of the percolation parameter, and the correlation between sites decays as distance to the power \alpha = d-2. We will discuss two results. First, in dimensions d \ge 4, we show that the number of clusters is asymptotically Gaussian for all non-critical levels, just as in the i.i.d. case. Second, in dimension d=3, we show that the limit law may be Gaussian or non-Gaussian, depending on the level. This difference turns out to reflect the fact that the correlation decay exponent \alpha = d-2 is less than d/2 if and only if d = 3, and we conjecture that the same limits appear for all correlated percolation models with decay exponent \alpha \in [d/3,d/2).

The talk is based on joint work with Michael McAuley (TU Dublin).

Michael McAuley (TU Dublin): Geometric functionals of smooth Gaussian fields

31 July

Smooth Gaussian fields are widely used for modelling phenomena across scientific disciplines (e.g. in cosmology, medical imaging, quantum chaos and machine learning). In many of these areas, statistical analysis can be naturally related to the geometric properties of the field. Understanding these geometric properties has, in turn, motivated deep theoretical questions which require techniques from many different areas (e.g. differential/integral geometry, analysis, probability theory and mathematical physics) to answer.

In this talk I will give an overview of this research topic and touch on some recent progress in proving limit theorems (i.e. laws of large numbers and central limit theorems) for geometric functionals associated with Gaussian fields.

Based on joint work with Dmitry Beliaev (University of Oxford) and Stephen Muirhead (University of Melbourne)

Roxanne He (Melbourne): Cutoff for the SIS model with self-infection and mixing time for the Curie-Weiss-Potts model

7 August

In this talk, I present joint work with Malwina Luczak and Nathan Ross, where we study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional “self-infection” is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we show that it exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. At the end, I also briefly discuss joint work with Jackie Lok on the mixing time for the restricted Curie-Weiss-Potts model in the subcritical temperature regime, along with an application.

Maximilian Nitzschner (Hong Kong UST): Bulk deviation lower bounds for the simple random walk

14 August

In this talk we present large deviation lower bounds for the probability of certain bulk-deviation events depending on the occupation-time field of a simple random walk on the Euclidean lattice in dimensions larger or equal to three. As a particular application, these bounds imply an exact leading order decay rate for the probability of the event that a simple random walk covers a substantial fraction of a macroscopic body, when combined with a corresponding upper bound previously obtained by Sznitman. As a pivotal tool for deriving such optimal lower bounds, we recall the model of tilted walks which was first introduced by Li in order to develop similar large deviation lower bounds for the probability of disconnecting a macroscopic body from an enclosing box by the trace of a simple random walk. We then discuss a refined local coupling with the model of random interlacements which is used to locally approximate the occupation times of the tilted walk. Based on joint work with A. Chiarini (University of Padova).

Nathan Ross (Melbourne): Gaussian random field approximation via Stein’s method, with applications to wide random neural networks

21 August

We describe a general technique to derive bounds for Gaussian random field approximation with respect to a Wasserstein transport distance in function space, equipped with the supremum metric. The technique combines Stein’s method and infinite dimensional Gaussian smoothing, and we apply it to derive bounds on Gaussian approximations of wide random neural networks of any depth. The bounds are explicit in the widths and natural parameters of the neural network. The talk covers joint works with Krishnakumar Balasubramanian, Larry Goldstein, and Adil Salim; and A.D. Barbour and Guangqu Zheng.

Adrian Röllin (NU Singapore): When Markov processes collapse, and what to do about it

28 August

We consider sequences of Markov processes that exhibit increasingly strong drifts towards a subspace of the state space and therefore “collapse” in the limit. The mathematical challenge is to prove process-level convergence, since the generators of the processes “blow up” outside the subspace. We show that Lyapunov functions for Markov processes and the Meyer-Zheng topology are convenient tools in such situations. We provide two examples — the first is a Moran model in a homogeneously mixing population with random resampling rates and the second is a voter model on a dynamically evolving random graph. This is joint work with Siva Athreya and Frank den Hollander.

Jesse Goodman (U Auckland): Saddlepoint approximations for likelihoods

4 September

The saddlepoint approximation is a systematic method for converting a known generating function into an approximation for an unknown density function. Interpreted instead as an approximation to the unknown likelihood function, the saddlepoint approximation can be maximized to compute the saddlepoint MLE for a given observed value. This talk will explain how the saddlepoint approximation can be interpreted with a statistical lens, and describe a class of models with theoretical guarantees for the effect of using the saddlepoint MLE as a substitute for the unknown true MLE. The talk will also demonstrate new tools to simplify and automate the computation of saddlepoint MLEs and to quantitatively assess the amount of approximation error. Based on joint work with Godrick Oketch and Rachel Fewster.

Nick Beaton (Melbourne): Chemical distance for the half-orthant model

18 September

The half-orthant model is a partially oriented model of a random medium involving a parameter $p\in [0,1]$, for which there is a critical value $p_c(d)$ (depending on the dimension $d$) below which every point is reachable from the origin. We prove a limit theorem for the graph-distance (or "chemical distance") for this model when $p<p_c(2)$, and also when $1-p$ is larger than the critical parameter for site percolation in $\mathbb{Z}^d$. The proof involves an application of the subadditive ergodic theorem. Novel arguments herein include the method of proving that the expected number of steps to reach any given point is finite, as well as an argument that is used to show that the shape is "non-trivial" in certain directions. This is joint work with Mark Holmes and Xin Huang. (2)$,>

Rongfeng Sun (NU Singapore): The Critical 2D Stochastic Heat Flow: disordered system meets singular SPDE

25 September

We discuss recent progress in the study of the 2-dimensional stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation, which are critical singular stochastic partial differential equations (SPDEs) that lie beyond existing solution theories. Both the 2D SHE and KPZ undergo a phase transition, and the solution of the 2D SHE at the critical point leads to the so-called critical 2D stochastic heat flow. This provides a rare example of a model in the critical dimension and at the critical point with a non-Gaussian limit. Our approach is motivated by the scaling limits of disordered systems, in particular, the directed polymer model in random environment for which disorder is marginally relevant in 2D. Based on joint work with F. Caravenna and N. Zygouras.

Weijun Xu (Peking U) Belz Short Course: Ergodicity of diffusion processes

26 September, 2 October, 9 October

We discuss ergodicity problems for continuous time Markov processes, with a focus on diffusions. Along the way, we will naturally encounter some landmarks in the development of stochastic analysis, such as Ito's SDEs, Malliavin calculus, etc. We will also discuss what happens in infinite dimensional situations (SPDEs).

Tejas Iyer (Weierstrass Institute): Leadership in growth processes

2 October

Consider a model where N 'agents' possess 'values' subject to increase over time. More precisely, these values are represented by N_0 valued increasing processes, with random, independent waiting times between jumps. We show that the event that a single agent possesses the maximum value for all sufficiently large values of time (called 'leadership') occurs with probability zero or one, and provide necessary and sufficient conditions for this to occur. In the particular case when waiting times are mixtures of exponential distributions, we improve a well-established result on the 'balls in bins' model with feedback (a model also known as non-linear P\'olya urns), removing the requirement that the feedback function be bounded from below and also allowing random feedback functions. The result is closely related to results regarding series of independent random variables. During the talk we will elucidate this connection, and then, time permitted, outline the technicalities associated with the proofs.

arXiv ref: https://arxiv.org/abs/2408.11516

Mihai Gradinaru (Rennes 1): Lévy driven non-linear Langevin type equations

16 October

We will consider a one-dimensional kinetic stochastic model driven by a stable Lévy process, with a non-linear time-inhomogeneous drift. More precisely, a process $(v,x)$ is considered, where $x$ is the position of the particle and its velocity $v$ is the solution of a stochastic differential equation with a drift of the form $t^{-\gamma}F(v)$. The behaviour of the process $(v,x)$ will be described when the noise is small or with a fixed noise in large time.

Xi Geng (Melbourne): Long time asymptotics for the parabolic Anderson model in the hyperbolic space

23 October

In this talk, we establish the exact second-order moment asymptotics for the parabolic Anderson model in the hyperbolic space with a time-independent, regular, isometry-invariant Gaussian potential. Although the solution is defined under hyperbolic geometry, surprisingly it turns out that the fluctuation exponent is determined by an Euclidean variational problem which is insensitive to the underlying geometry. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics. On the other hand, the almost-sure asymptotics becomes drastically different from the Euclidean case due to the exponential volume growth in negative curvature.

This is based on a recent joint work with Weijun Xu (Peking University) as well as an ongoing project with Weijun and my PhD student Sheng Wang.

Illia Donhauzer (La Trobe): Superpositions of continuous autoregressive random fields (supCAR fields).

30 October

The talk will introduce the supCAR random fields that form a rich class of homogeneous and isotropic random fields constructed as superpositions of CAR (continuous autoregressive) random fields. The supCAR random fields possess infinitely divisible marginal distributions and flexible dependence structures (including long-range dependence). The talk will discuss the existence of the supCAR fields and their properties. Special emphasis will be given to new functional limit theorems for the supCAR fields and properties of the limit processes.

Joe Yukich (Lehigh) Belz Short Course: Asymptotic analysis of statistics of random geometric structures

15 January

Recent years have seen the development of tools leading to the limit theory of statistics of geometric structures built on possibly correlated spatial data, as the number of data points tends to infinity.  Malliavin-Stein methods, Poincare inequalities, and stabilization are the tools lying at the heart of these methods.

We introduce and survey these methods and show that they lead to laws of large numbers, variance asymptotics and central limit theorems for functionals of random graphs on point processes, random convex hulls, and random deposition models.

Yuzuru Inahama (Kyushu): Wong-Zakai approximation of density functions

6 March

In this talk we prove the Wong-Zakai approximation of probability density functions of solutions at a fixed time of rough differential equations driven by fractional Brownian rough path with Hurst parameter H (1/4<H≤1/2). Besides rough path theory, we use Hu-Watanabe’s approximation theorem in the framework of Watanabe’s distributional Malliavin calculus. When H=1/2, the random rough differential equations coincide with the corresponding Stratonovich-type stochastic differential equations. Even for that case, our main result seems new.

Renjie Feng (Sydney): Determinantal point processes on spheres: Multivariate linear statistics

13 March

I will talk about the multivariate linear statistics (also known as U-statistics) of determinantal point processes on unit spheres. I will first present a graphical representation for the cumulants of the multivariate linear statistics, extending the famous Soshnikov’s formula for the univariate case. Then I will explain how we derive the 1st and 2nd Wiener chaos using this graphical representation. We computed sphere cases as introductory examples, but the method can be applied to any other determinantal point processes. This is based on the joint work with F. Goetze and D. Yao.

Emma Horton (Warwick): Genealogies of branching Markov processes

20 March

Branching processes are pertinent to understanding many different real world processes such as cell division, population growth and neutron transport. In particular, understanding their genealogical structures can prove useful for parameter estimation, Monte Carlo simulations and scaling limits. In this talk we discuss a decomposition of the branching process known as the many-to-few formula, which allows one to understand the behaviour of a branching process in terms of a weighted subtree. I will then give two applications of this decomposition to demonstrate its use in understanding the genealogical structure of the branching process.

Vincent Liang (Melbourne): On boundary crossing probabilities of diffusion processes

27 March

We discuss two results related to the probability $F(g_-,g_+)$ that a general time-inhomogeneous diffusion process $X$ stays between two curvilinear boundaries $g_-$ and $g_+$ (possibly with $g_{\pm} = \pm \infty$) during a finite time interval. Joint work with K. Borovkov.

Nadia Sidorova (UCL): Edge-reinforced branching random walk on the triangle

10 April

Edge-reinforced random walk (ERRW) is a random process on the vertices of a graph that is more likely to cross the edges it has visited in the past. Depending on the strength of the reinforcement, one-dimensional ERRW can either exhibit localisation (eventually moving back and forth across a single edge) or remain transient. We consider a model where a single ERRW is replaced by an exponentially growing number of random particles, and we study its localisation properties on the simple triangle graph. Using the dynamical systems approach we analyse the frequencies with which the edges are traversed and prove their almost sure convergence. We discuss the scenarios when those frequencies become negligible for one or two edges (dominance). We also discuss the situation when an edge stops being traversed entirely (monopoly). This is a joint work with Giordano Giambartolomei.

Konstantin Borovkov (Melbourne): Large deviation probabilities for random walks: Light vs heavy tails

17 April

Random walks are important models for real-life processes, including claim surplus processes for insurance companies, where light-tailed jump distributions are usually assumed for life insurance, and heavy-tailed – for non-life one. We will present the fundamentals of the large deviations theory for random walks, touching on both light- and heavy-tailed cases, and outline results on the asymptotics of the probabilities of remote curvilinear boundary crossing by the walks. We will also discuss large deviation results in the case of right-censored jumps in the random walk. Part of the work was joint with A. Chong.

Greg Markowsky (Monash): Ways in which the geometry of plane domains is reflected in the distribution of Brownian motion exit times

24 April

The distribution of the exit time of Brownian motion from a plane domain carries a great deal of information about the shape and size of the domain. There are beautiful connections between this fact and classical complex analysis. I will present some recent results on this, and discuss some open problems and conjectures.

Phillip Yam (City U Hong Kong): Mean Field Games, their FBSDEs and Master Equations

1 May

Modeling collective behaviors of individuals in account of their mutual interactions arisen in various physical or sociological dynamical systems have been one of the major problems in the history of mankind. To resolve this matter, a completely different macroscopic approach inspired from statistical physics had been gradually developed in the last decade, which eventually leads to the primitive notion of mean field game theory. In this talk, we shall introduce a theory of global-in-time well-posedness fora general class of mean field game problems, which include as an example setting with quasi-convex payoff functions as long as the mean field sensitivity is not too large.

Aram Perez (Monash): Multivariate Non-Normal Stein’s Method with Application to the Critical, Mean-Field O(N) Model

8 May

Stein’s Method is a powerful tool for studying distances between distributions of random variables. It has been used widely in the context of statistical mechanics to obtain approximations for various thermodynamic quantities. We will discuss an application of Stein’s Method to the O(N) model, a model of magnetism, and show how recent advances in Stein’s theory allow us to study the limiting behaviour of the magnetisation at the phase transition.

Kazutoshi Yamakazi (Queensland): A series expansion formula of the scale matrix

14 May

We present a new series expansion formula for the scale matrix of Markov additive processes with a constant drift and general finite-activity one-sided jumps. This formula generalizes the series expansion formula for the scale function derived in Landriault & Willmot (Scand. Actuar. J., 2020) for the Cramer-Lundberg process. We discuss its applications in ruin theory and sequential testing. This work is a collaboration with J. Ivanovs (Aarhus University).

Mario Kieburg (Melbourne): Stable and Group Invariant Hermitian Random Matrices

22 May

The classification and characterisation of stable random variables is one of the central questions in classical probability theory. Starting with the classical central limit theorem for Gaussian random variables, it was soon extended by Lévy to α-stable distributions for univariate statistics. Those results were generalised to vectors by Rvačeva which completed the multi-variate case. As flat symmetric matrix spaces are vector spaces, one could think that the problem is solved for those, as well. However, matrices give rise to other intriguing quantities such eigenvalues, singular values and eigenvectors whose statistics have a non-trivial relation to the matrix entries. In a series of works, Jiyuan Zhang and I have studied the classification of joint probability densities of the eigenvalues for unitarily invariant Hermitian random that are stable with respect to matrix addition. We have analysed their domain of attraction and the optimal rate of convergence when generating those from a stable random vector. In the talk, I will report about these results and what the future directions of research will be.

Simon Harris (Auckland): Universal genealogies of samples from Galton-Watson trees with heavy-tailed offspring

29 May

Consider some population evolving stochastically in time. Conditional on the population surviving until some large time T, take a sample of particles from those alive. What does the ancestral tree drawn out by this sample look like? Some special cases were known, e.g. Durrett (1978), O’Connell (1991), but we will discuss an approach behind some more recent advances for Bienyame-Galton-Watson (BGW) processes conditioned to survive. In near-critical settings with finite offspring variance, the same universal limiting sample genealogy always appears up to some deterministic time change. This genealogical tree has the same binary tree topology as a Kingman coalescent, but where the coalescent (or split) times can be represented as a mixture of IID times - this very roughly interpreted as a mixture of time changed ‘slowed down’ Kingman coalescents. In contrast, in critical infinite variance offspring settings, we find that more complex universal limiting sample genealogies emerge that exhibit multiple-mergers, these being driven by massive birth events within the underlying population. The key tool in our proofs is a change of measure involving k distinguished particles, also known as spines, which corresponds to k-size biasing and discounting by the population size. Some ongoing work and open problems will also be mentioned. This talk is based on work in collaboration with M.Roberts (Bath), S.Johnston (KCL) in AAP (2020), with J.C.Pardo (CIMAT), S.Johnston in AOP (2024), and with S.Palau (UNAM), J.C.Pardo (2022+).

Allan Sly (Princeton): Transience for the interchange process in dimension 5

12 June

The interchange process $\sigma_T$ is a random permutation valued stochastic process on a graph evolving in time by transpositions on its edges at rate 1. On $Z^d$, when $T$ is small all the cycles of the permutation $\sigma_T$ are finite almost surely. In dimension $d \geq 3$ Toth conjectured that infinite cycles appear when $T$ is large. The cycles can be interpreted as a random walk which interacts with its past and we give a multi-scale proof establishing transience of the walk (and hence infinite cycles) when $d\geq 5$. In a finite volume we establish Poisson-Dirichlet statistics for the largest cycles.