Gallery

Here are some pictures of the random processes that I’m interested in. Feel free to use any of them but please give credit.

Hyperbolic BBM

This is from an ongoing project on branching Brownian motion in hyperbolic space, modelled via the Poincaré disk model. Here, a single Brownian particle converges to a uniform point on the boundary. For BBM, the empirical measure converges to a random measure on the boundary. If we increase the branching rate, then this random measure becomes more spread out in an appropriate sense. In the picture, the branching rates left to right are $0.12, 0.4$ and $1$.

Simulations of climate distance

This is from a project where I have helped a group of ecologists studying climate distance: for a given location on the planet, given the predicitions for climate change, how far away do you have to move to find a location that will have the same climate in the future as the initial location does now? This is one way to measure the speed (rather than the severity) of climate change. Analysing this helps to understand which areas will act as refuges for biodiversity. Here I simulate this for a mountain range adjacent to some plains, this shows that the climate distance will be lower for locations on the slope of the mountain compared to the plains.

BBM with space inhomogeneous branching rate

This is from the paper where we consider branching Brownian motion in two dimensions where the branching rate depends on the direction. The branching rate is maximised in the direction of the $x$-axis. In the picture, the different colours correspond to different choices of a parameter that controls how much smaller the branching is in directions other than the $x$-axis. Observe that the shape of the blob becomes less spherical for the smaller blobs where the inhomogeneity is more pronounced.

Here are some realisations in different colours sibBBM_2 and sibBBM_3.

Branching random walk on random trees

Here we consider a branching random walk that takes place on a random tree. The random tree can be seen as a model for a random environment in very high dimension. As the process grows, it explores the tree. Here we plot the trace of the branching random walk. This is a figure from our paper here arXiv [2502.07363].

The colours illustrate the distance from the root of the tree where the branching random walk is initiated. Here are some more realisations brw_2, brw_3, brw_4.

Marchal’s tree growth and stable trees

Marchal’s tree growth is a tree-valued Markov chain. Once rescaled properly, the trees converge as metric spaces. The limiting objects are called stable trees. Here are some simulations of Marchal’s tree growth after 25000 steps for different values of alpha, they approximate stable trees. Here, alpha takes values in 2, 1.9, 1.6 and 1.4. When the parameter alpha is lowered the degree of some vertices in the tree becomes bigger and bigger leading to visual accumulation points. In the case alpha = 2, the stable tree is also called the Brownian continuum random tree (BCRT).

You can download vectorised versions here: 1.4 stable tree, 1.6 stable tree, 1.9 stable tree, BCRT. Also, here is a GIF of a breadth first search of one of those trees.

Dynamic Widom-Rowlinson model

The Widom-Rowlinson model is a stochastic model for liquid-vapour transitions. Mathematically, it is a point process defined on the plane of points of two different colours. Each point comes with a disk of diameter one, points of different colour must not overlap whereas points of the same colour ignore each other. Projecting on one colour yields the model for the liquid-vapour transition. This (random) model comes with a stochastic dynamic where points die at rate 1 and are born at rate z as long the birth is compatible with the existent configuration.

I don’t research on this model myself but I learned about it in a course of Frank den Hollander at the PIMS summer school 2022.

Here are some GIFs of the dynamics, (large files, ~20MB each): intensity 0.025, intensity 0.05, intensity 0.075, intensity 0.1.