Probability is a beautiful and ubiquitous field of modern mathematics that can be loosely described as the mathematics of uncertainty. It has applications in all areas of pure and applied science, and provides the theoretical basis for statistics. Many recent Fields Medallists have been recognised for their work in probability.

Faculty (and selected interests):

Andrew Barbour (Stein’s method, mathematical biology)

Nick Beaton    (statistical mechanics, combinatorics)

Howard Bondell   (statistics)

Kostya Borovkov (Markov chains, applications of stochastic processes in biology and finance)

Mark Fackrell (stochastic modelling, game theory)

Peter Forrester (random matrices, multivariate probability distributions)

Xi Geng      (stochastic analysis and rough path theory)

Jan de Gier   (statistical mechanics)

Tony Guttmann (combinatorics and statistical mechanics)

Sophie Hautphenne (branching processes, Markov population models)

Mark Holmes (random media, random walks, interacting particle systems, reinforcement processes)

Mario Kieburg (random Matrices, point processes, multivariate probability distributions, statistical mechanics)

Stephen Muirhead (random fields, random media, percolation)

Nathan Ross (Stein’s method, random graphs and networks)

Michael Stumpf (phylodynamics and lineage trees, random hypergraphs, stochastic dynamics of gene expression, non-equilibrium statistical mechanics of biological networks)

Peter Taylor (stochastic modelling, operations research)

Aihua Xia (point processes, applied probability, Stein’s method)

PhD Students:

Tim Banova, Mark Holmes (principal supervisor), Nick Beaton (co-supervisor)

Aaron Chong, Kostya Borovkov (principal supervisor), Nathan Ross (co-supervisor)

Roxanne He, Nathan (principal supervisor), Kostya Borovkov (co-supervisor)

Vincent Liang, Kostya Borovkov (principal supervisor), Aihua Xia (co-supervisor)

Duy Phat Nguyen, Kostya Borovkov (principal supervisor), Aihua Xia (co-supervisor)

Sheng Wang, Xi Geng (principal supervisor), Aihua Xia (co-supervisor)

Recently completed research students:

Tianshu Cong, PhD.   Convergence rate for statistics of point processes (Now at N.U. Singapore)

Xin Huang, MPhil.   A shape theorem for the half-orthant model and the features of its shape

Qingwei Liu, PhD.  On moderate deviations in Poisson approximation and its applications (Now at Nanyang Technological U.)

Tiffany Lo, PhD.  Limit theorems for some network growth models (Now at Uppsala U.)

Here are some regular online seminar series that may be of interest:

Asia-Pacific Seminar in Probability and Statistics

Probability Victoria online seminar

One world probability seminar

Open Online Probability School

• Feb 26: Peter Hall Lecture 2020

  Counting Self-Avoiding Walks on a Lattice, from Combinatorics to Physics     Hugo Duminil-Copin  

  25 February, 2020 Event
• 26 Feb: Probability Seminar

  Triviality of the 4D Ising Model    Hugo Duminil-Copin  

  25 February, 2020 Event, Probabil...
• Clusters of Brownian loops in dimensions greater than 3

  Wendelin Werner (ETH).

  18 December, 2019 Probability Sem...
• Random Fluctuations Around a Stable Limit Cycle in a Stochastic System with Parametric Forcing

  Cindy Greenwood (UBC).

  16 December, 2019 Probability Sem...
• Stability of the Elliptic Harnack Inequality

  Martin Barlow (UBC).

  2 December, 2019 Probability Sem...
• Local Colouring and Constraint Solving

  Nov-Dec 2019 - Ander Holroyd (UW)

  1 November, 2019 Mini Course

Previous probability seminars

2024

• Emma Horton (Warwick) March-April

2023

• Victor Kleptsyn (CNRS/Rennes) March

2022

• Emma Horton 2022-2023 (Postdoc)

2020

• Hugo Duminil-Copin (IHES) Feb 25-26

2019

• Wendelin Werner (ETH) Dec 18
• Cindy Greenwood (UBC) Dec 15 – Dec 30
• Martin Barlow (UBC) Dec 2
• Alejandro Ramirez (PUC) Nov 30 – Dec 4
• Ander Holroyd (UW) Nov 25 – Dec 21
• Han Gan (Waikato) 23 Sept – 23 Dec
• Tom Salisbury (York) 4 Feb – 20 Feb

2018

• Ruth Williams (UC San Diego) 3 Dec – 18 Dec
• Ivan Corwin (Colombia) 1 Nov – 7 Nov
• Denis Denisov (Manchester) 25 Sept – 26 Oct
• Carmen Minuesa Abril (U. Extremadura) 15 Sept  – 14 Dec
• David Croydon (Kyoto) 6 Aug – 31 Aug
• Adrian Röllin (NU Singapore) 16 July – 20 July
• Victor Kleptsyn (Rennes) 11 Apr – 10 May
• Ilze Ziedins (U Auckland) 26 Mar – 6 Apr
• Sidney Resnick (Cornell) 12 Mar – 14 Mar
• Peter Jagers (Chalmers UT and Gothenburg U) 5 Mar – 12 Mar
• Tom Salisbury (York U) 19 Feb – 2 Mar
• Ellen Powell (ETH Zürich) 7 Jan – 9 Jan

2017

Polya’s urn    in this model, starting with one red ball and one black ball in an urn, we repeatedly select a ball at random from the urn and return it, while also adding another ball of the same colour.  The probability of selecting a red ball is proportional to the number of red balls raised to the power alpha.  Observe the different behaviour as we vary alpha.  There is a phase transition when alpha=1.

WARM process on a graph    at each step of this process we choose a vertex at random, and from the chosen vertex we choose one of the edges (colours) with probability proportional to the number of times (plus 1) that the edge has previously been selected, raised to the power alpha.  Now look at the set of edges that are chosen infinitely often.  There are in fact infinitely many phase transitions for this model!  The randomise button allows us to change the vertex selecction probabilities from uniform to “random”.

Exclusion process on a circle  in this model, particles try to move around a circle, with a given probability of trying to go clockwise.  When a particle tries to make a move, it fails if there is already another particle in the spot that it wanted to move to.  How fast does each particle move around the circle when there are many particles?

The voter model  in this model, each site carries a different vote (colour).  Some of the voters may be obstinate (they will never change their vote) – you can set these vertices by clicking on them – while others are influenced by their neighbours.  If none of the voters are obstinate (or all obstinate voters are of the same colour), then eventually all voters will agree.

Research in our group covers a diverse range of theoretical and applied probability and stochastic processes, including: stochastic approximation, the theory of queues and stochastic networks, random walks, random graphs and combinatorial structures, reinforcement processes, interacting particle systems, stochastic dynamical systems.

Recent examples include:

M Kieburg, SH Li, J Zhang, and PJ Forrester. Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics. Constructive Approximation, 1-46 (2023).

A.D. Barbour, G. Brightwell and M. J. Luczak. Long-term concentration of measure and cut-off. Stochastic Processes and their Applications 152, 378-423 (2022).

Beaton, N. and Holmes, M. The mean square displacement of random walk on the Manhattan lattice. Statistics and Probability Letters, Volume 193, (2023), 109706

P. Taylor. Spatial queues with a moving server. Queueing Systems, vol.100,no.3-4,, pp. 261-263, (2022).

D. Bertacchi, P. Braunsteins, S. Hautphenne, and F. Zucca.
Extinction probabilities in branching processes with countably many types: a general framework. ALEA, Lat. Am. J. Probab. Math. Stat., 19:311-338, (2022).

Holmes, M. and Salisbury, T. A shape theorem for the orthant model. Annals of Probability 49:3, 1237-1256, (2021)

D. Beliaev, S. Muirhead and A. Rivera. A covariance formula for topological events of smooth Gaussian fields. Ann. Probab. 48(6) (2020), 2845–2893.

D. Beliaev, M. McAuley and S. Muirhead . On the number of excursion sets of planar Gaussian fields. Probab. Theory Related Fields 178(3–4) (2020), 655–698

Borovkov, K. and He, P. Limit theorems for record indicators in threshold Fα-schemes. To appear in Theory Probab. Appl. (2020) arXiv:1903.03753

Hirsch, C., Holmes, M. , and Kleptsyn, V. Absence of WARM percolation in the very strong reinforcement regime. Ann. Appl. Probab. 31 (1), 199-217, (2021).

YP Förster, M Kieburg and H Kösters. Polynomial ensembles and Pólya frequency functions. J. Theoretical Probability 34 (4), 1917-1950, (2021).

Bostan, A., Elvey-Price, A., Guttmann, A.J., and Maillard, J.-M.  Stieltjes moment sequences for pattern-avoiding permutations Electronic J. of Combinatorics Volume 27, Issue 4 (2020).

de Gier, J., Kenyon, R., and Watson, S.S.  Limit shapes for the asymmetric five vertex model. Communications in Mathematical Physics volume 385, pages 793–836 (2021).

Barbour, A., and Xia, A. Multivariate approximation in total variation using local dependence. Electr. J. Prob. 24 (2019), paper no. 27, 1-35.

Barbour, A. and Rollin, A. Central limit theorems in the configuration model.  Ann. Appl. Probab. 29, 1046-1069, 2019.

Reinert, G. and Ross, N. Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphs. Ann. Appl. Probab. 29 (5), 3201-3229 (2019).

Holmes, M. and Perkins, E.  On the range of lattice models in high dimensions.  Prob. Th. Related Fields. (2020).

P. Braunsteins and S. Hautphenne. Extinction probabilities of Lower Hessenberg branching processes with countably many types. Annals of Applied Probability, 2019, 29(5):2782-2818.

Geng, X. and Boedihardjo, H. Tail asymptotics of the Brownian signature.  Trans. Amer. Math. Soc.: 585-614, 2019.

Beaton, N.R., Eng, J.W. and Soteros, C.E.  Knotting statistics for polygons in lattice tubes
Journal of Physics A: Mathematical and Theoretical 52, (2019), 14403.

Borovkov, K. Gaussian process approximations for multicolor Pólya urn models. J. of Applied Probability, 58(1), 274-286. (2020)

Bean, N. G., Latouche G., and  Taylor P. Physical Interpretations for Quasi-Birth-and-Death Process Algorithms. Queueing Models and Service Management. 1(2), 59-78, 2018.

Chen, Z., de Gier, J., Hiki, I.  and Sasamoto, T.  Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process. Phys. Rev. Lett. 120, 2018.

Probability is a beautiful mathematical subject in its own right.  Students who are interested in diverse subjects such as statistics, statistical mechanics, mathematical biology, mathematical finance, analysis, discrete mathematics, optimisation….  are also likely to benefit from taking subjects in probability.

The following U. Melbourne subjects are strongly recommended for students who are interested in gaining expertise in probability:

• Advanced Calculus (MAST10021)
• Advanced Linear Algebra (MAST10022)
• Probability (MAST20004)
• Real Analysis (MAST20026)
• Probability for Inference (MAST30020)
• Stochastic Modelling (MAST30001)
• Metric and Hilbert Spaces (MAST30026)

• Measure Theory (MAST90012)
• Advanced Probability (MAST90081)
• Stochastic Calculus with Applications (MAST90059)
• Advanced Topics in Stochastic Models (MAST90112)
• Random Processes (MAST90019)

Other subjects that are (sometimes) offered at U. Melbourne that are very useful for probability students to know about:

• Methods of Mathematical Physics (MAST30031)
• Graph Theory (MAST30011)
• Complex Analysis (MAST30021)

• Mathematical Statistical Mechanics (MAST90060)
• Functional Analysis (MAST90020)
• Enumerative Combinatorics (MAST90031)
• Riemann Surfaces and Complex Analysis (MAST90056)
• Exactly Solvable Models (MAST90065)
• Mathematical Statistics (MAST90082)
• Random Matrix Theory (MAST90103)
• Advanced Mathematical Statistics (MAST90123)

Tony Guttmann has been awarded the title of Member of the Order of Australia, for significant service to  the mathematical sciences, and to education.

Peter Taylor received the 2019 ANZIAM Medal

Nick Beaton, Jan de Gier, Tony Guttmann and coauthors received the 2018 Gavin Brown Prize

Peter Taylor received the 2018 George Szekeres Medal

The Joker Paradox

Suppose that Batman and the Joker walk on the vertices of a “bipartite graph” (e.g. the corners of a square), and Batman catches the Joker when they are at the same vertex at the same time (they move independently of each other until the Joker is caught).  They start from different vertices of the same parity (e.g. opposite corners of the square), and Batman always moves to one of the neighbouring vertices at random.  The Joker survives for twice as long on average if he is infinitesimally lazy (i.e. doesn’t move on any given step with infinitesimally small but positive probability) than if he is never lazy (i.e. always moves to one of the neighbouring vertices).

(See “A paradox for expected hitting times” by Holmes and Taylor)

A paradox for Polya’s urn

Starting with 1 black ball and 1 red ball in an urn:  Choose a ball at random from the urn.  If it is black, put it back in the urn and add another black ball.  Now continue the above (adding a black ball each time you select black), and stop when you choose a red ball.

It turns out that the random time T until you select a red ball is finite, but the average value of T is infinite.

Now do the same experiment above, except that we start with 10,000,000 black balls and 2 red balls in the urn.  It turns out that the average time that it takes you to select a red ball is now finite!