Probability in Seattle

Research gallery
Below we include a gallery of pictures representing a small sample of probability research in Seattle. For a more comprehensive view of this research, see individual homepages listed under People and the lists of Publications.
Robin problem in fractal domains

There is a deep and fruitful interplay between probability theory and analysis. For example, stochastic analysis can be used to study physics or physiology of fractal sets, such as human lungs. The flow of oxygen through the semi-permeable lung membrane can be modeled using the heat equation with Robin boundary conditions (also known as the third boundary problem). For Robin problem in fractal domains, see the paper by Bass, Burdzy and Chen. Probabilistic methods are also very effective in studying heat propagation by jump-diffusions in non-homogeneous media. See the preprints page of Zhenqing Chen.

Stable allocations

Each centre simultaneously starts growing a circle, all at the same speed. A centre claims all territory encountered by its circle (unless another centre has claimed it earlier), until the centre reaches its fair share of the area. Suppose that points "prefer" to be in a territory of a nearby centre, while centres "prefer" to have a nearby territory. Call a partition "stable" if there do not exist a site and centre which are not allocated to each other, but both prefer each other compared with the current partition. The construction above yields the only stable partition. This notion of stability was introduced by Gale and Shapley in a 1962 paper, "College Admissions and the Stability of Marriage." We adapt the algorithm they invented for those problems to our setting. (Based on research of C. Hoffman, A. Holroyd, Y. Peres).

Harmonic measure

Curve (blue) with least harmonic measure at 0 that meets every ray from 0 to the unit circle. The probability that a Brownian traveler (starting at the origin) hits the green curve before hitting the blue curve can be estimated by counting boxes: 12 out of 512 of the outer most boxes meet the green curve. The actual probability is quite close: 0.02287... The picture was drawn using the Zipper algorithm for conformal mapping. (Based on research of Don Marshall and Carl Sundberg.)

Card shuffling and Diophantine approximation

The ``overlapping-cycles shuffle'' mixes a deck of n cards by moving either the nth card or the (n-k)th card to the top of the deck, with probability half each. The figure shows the inverse spectral gap for the location of a single card, which, as a function of k and n, has surprising behavior. For example, suppose k is the closest integer to α n for a fixed real α in (0,1). For rational α the spectral gap is Θ(n-2), while for poorly approximable irrational numbers α, such as the reciprocal of the golden ratio, the spectral gap is Θ(n-3/2). See paper by Angel, Peres, and Wilson.

Hot spots and couplings

Suppose that a perfectly insulated body was unevenly heated. The `hot spots' conjecture asserts that after a long time, the hottest and coldest spots in the body will be on the boundary. This is not so for the planar tubular body illustrated in the picture. The hottest and coldest spots will be inside the body, as indicated by the arrows. This purely analytic statement was proved using `couplings', a probabilistic technique. The `hot spots' problem is still open for triangles with acute angles. For more information, see this page maintained by Chris Burdzy.

Gravitational allocation to Poisson points

Imagine stars at random locations in space (distributed according to a Poisson process). The gravitational force field of these stars partitions space into domains of attraction which have equal volume. In three and higher dimensions, Chatterjee, Peled, Peres and Romik (2006) established an exponential tail for the diameters of these domains, See ArXiv preprint. (Picture courtesy of M. Krishnapur).

Groves of trees

This figure shows a uniformly random grove, that is, a collection of trees such that each tree contains at least one of the marked nodes on the outer face. The partition on the nodes induced by this grove is 1|278|345|6. What is the probability distribution on partitions induced by uniformly random groves? It turns out that this distribution is determined by the electrical resistances between the nodes when we view the graph as an electrical network, and otherwise the internal structure of the graph does not matter. The formulas for these probabilities generalize Kirchhoff's formula for the resistance of a graph. There are analogous formulas for the pairings of chains in the double-dimer model and contour lines in the Gaussian free field. These formulas are relevant to multichordal versions of SLE2, SLE4, and SLE8. See paper by Kenyon and Wilson.

Red-green-blue model and conformal invariance

This picture illustrates the red-green-blue model from statistical physics, which is a sytem of loops obtained by superimposing three dimer coverings on offset hexagonal lattices. Here the red-green-blue configuration was generated within a triangular region, and then conformally mapped to the disk. One consequence of conformal invariance is that the loops within the disk are rotationally invariant in the limit of small lattice spacing. These red-green-blue loops appear to be closely related to SLE4 and double-dimer loops, but are more tightly nested than the double-dimer loops. See paper by Wilson.

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