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. Four of the last twelve 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)
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)
Stephen Muirhead (random fields, random media, percolation)
Nathan Ross (Stein’s method, random graphs and networks)
Peter Taylor (stochastic modelling, operations research)
Aihua Xia (point processes, applied probability, Stein’s method)
Tim Banova (probability theory)
Xin Huang (probability theory)
Here are some regular online seminar series that may be of interest:
Feb 26: Peter Hall Lecture 2020
Counting Self-Avoiding Walks on a Lattice, from Combinatorics to Physics Hugo Duminil-Copin25 February, 2020 Event
26 Feb: Probability Seminar
Triviality of the 4D Ising Model Hugo Duminil-Copin25 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
- Hugo Duminil-Copin (IHES) Feb 25-26
- 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
- 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
- Omer Angel (UBC) 7 Dec – 21 Dec
- Alejandro Ramírez (PUC Chile) 7 Dec – 16 Dec
- Edwin Perkins (U British Columbia) 17 Nov – 3 Dec
- Michel Mandjes 10 Oct – 20 Nov
- Dominic Schuhmacher (Göttingen) 20 Sept – 6 Oct
- Christian Hirsch (LMU Munich) 9 Apr – 12 May
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:
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).
Bostan, A., Elvey-Price, A., Guttmann, A.J., and Maillard, J.-M. Stieltjes moment sequences for pattern-avoiding permutations arxiv:2001.00393
de Gier, J., Kenyon, R., and Watson, S.S. Limit shapes for the asymmetric five vertex model. arXiv:1812.11934
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. arxiv: 1912.09665. (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.
Barbour, A., McVinish, R., and Pollett, P. K. Local approximation of a metapopulation’s equilibrium. J. Math. Biology 77, 765-793, 2018.
S. Hautphenne , M. Massaro, and P. G. Taylor. How old is this bird? The age distribution under some phase sampling schemes. Journal of Mathematical Biology, 2017, 75(6-7):1319–1347.
Holmes, M., and Salisbury, T.S. Ballisticity and invariance principle for random walk in non-elliptic random environment. Electron. J. Probab. Volume 22, paper no. 81, 2017.
Beaton, N., Bousquet-Mélou, M. de Gier, J., Duminil-Copin, H. and Guttmann, A. The critical fugacity for surface adsorption of SAW on the honeycomb lattice is 1+√2. Comm. Math. Phys. 326, 727–754, 2014.
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)
- 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!