[Picture credits: Nicolas Curien, Henry Jackson, Jason Miller]
The geometry of random surface has in recent years emerged as a subject of central importance in probability. This is a subject with strong links to conformally invariant random processes such as SLE and the Gaussian free field, the geometry of random planar maps, Gaussian multiplicative chaos, and what the physicists call Liouville quantum gravity.
On the left is a representation of large random planar map, embedded into 3-space. A circle packing representation of such a map is shown on the right. The central image is a piece of a simulation of Liouville Brownian motion, which is the canonical diffusion process on the limiting continuous random surface. A beautiful video by Henry Jackson, showing a simulation of Liouville Brownian motion (in a GFF landscape), can be found here: here.
Condensation of random walks and Wulff crystals
These pictures are simulations of a random walk in two dimensions which has been penalised by a factor , where is the range of the walk at time , is the (vertex) boundary of the range, and is a positive factor, to be thought of as an inverse temperature. As increases, the distribution favours increasingly condensed configurations. In this paper, it is proved that the diameter of the shape in two dimensions is . (Sharp bounds on the volume are also obtained in all dimensions).
This represents the configuration of a polymer in a poor solvent, and suggests a construction of the Wulff crystal using random walks, as opposed to the Ising model or percolation as has been the case up to now.
The existence of a limiting shape is an intriguing open problem. As tends to infinity, we believe that this shape converges (after normalisation) to a diamond.
Brunet-Derrida particle systems
Brunet-Derrida particle systems are a model for natural selection in a population dynamics. The population consists of individuals, represented by a cloud of particles on the real axis. The position of an individual represents its fitness, ie, chances of reproductive success. At each generation, every individual gives birth to two offsprings. Of these individuals, only the fittest (ie, rightmost) are retained to form the next generation (picture on left).
This is strongly related to a model of branching Brownian motion where individuals are killed if they touch a killing wall moving at speed to the right, where has to be chosen appropriately to keep the population size fixed roughly equal to . In this paper, it was shown that the right choice is . Furthermore, with this choice the genealogy of a sample in this population converges, after scaling time by , to the Bolthausen-Sznitman coalescent. This confirms, for this model, a host of nonrigorous predictions made by Brunet, Derrida and coauthors.
The Bolthausen-Sznitman coalescent (picture on right) is a universal object which describes, among other things, the limiting structure of certain mean-field spin glass models such as the Sherrington-Kirckpatrick model. This is probably not a coincidence as the Brunet-Derrida particle system can be rephrased as another version of such spin glass models. Many open problems remain to be understood in this direction
Multidimensional Brunet-Derrida particle systems
The system above has an interesting multidimensional generalisation, introduced in this paper with Lee Zhuo Zhao. Here the particles live in . We are given a score function which serves to rank particles. At each generation, individuals produce two offsprings. Of these individuals, only the fittest (ie, with highest score) are retained to form the next generation.
In the movie on the left is an example where and is the Euclidean norm. It is proved in this same paper that a cloud of particles eventually forms and travels in a random direction at positive (explicit) speed. The next picture shows a zoom of this cloud of particles. Interestingly, the cloud is spread out in direction orthogonal to motion. We believe this is a quite general feature of the model (eg, it should not be too sensitive to the choice of score function, subject to some natural assumptions.) It was proved in the same paper that, in the linear case , the cloud of particles travels in the direction ultimately. Furthermore, the spread of this cloud of particles is in the direction parallel to motion but at least in the direction orthogonal to motion. We believe this exponent is sharp.
Once the cloud of particles settles in an essentially one-dimensional geometry, we believe that the same conjectures as in the case hold: in particular, we also expect that the genealogy is ultimately governed by the Bolthausen-Sznitman coalescent, in the limit where .
The next picture is a simulation of this process where , where is the location of the centre of mass (its trajectory is depicted in yellow on the picture). This particular choice was suggested by Jeremy Quastel, who also suggested the name ``Brownian bees". The yellow circle on the picture is the circle of radius , where is the th zero of the Bessel function of the first kind . We conjecture that in the limit where , particles are confined within this disc. The movie below is a big file and may take a while to download but is well worth seeing.
Random walks on
Random walks on the permutation group have some fascinating behaviours. On the left, is the Cayley graph of generated by all transpositions, while on the right only by adjacent transpositions. In the former case, it is proved in this paper that the distance to the starting point of a random walk undergoes a phase transition after steps. This is related to changes in the geometric structure of the graph: see this paper for hyperbolic properties in the sense of Gromov.
Bulding on these ideas, I obtained in this paper, with Oded Schramm and Ofer Zeitouni, a probabilistic proof of the celebrated Diaconis-Shahshahani result that the mixing time of random transpositions occurs at time . Furthermore, the proof extends with little extra work to random walks on where the step distribution is uniform over a given conjugacy class. In particular the paper handled the case of cycles, solving an old conjecture raised by the work of Diaconis and Shahshahani.