MAPPING STOCHASTIC PROCESSES
Intriguing mappings exist between various stochastic processes . To give a flavour of this, consider a Galton-Watson branching process with geometric offspring distribution and a random walk excursion. The mapping proceeds as follows: starting with the root progenitor of the branching process, perform a depth-first search (also known as a Harris walk) by ascending the generations and choosing the left-most child whenever possible. Upon arriving at an individual with no children, retrace the path back to the last child overlooked, and resume the search. Eventually, the path will visit all children and return back to the root progenitor (red path, left figure). If during this depth-first search one records the generation number from the root, one obtains an excursion path (right figure). If the offspring distribution of the branching process is geometric, then it can be shown that the excursion path is a simple random walk. To unambiguously define the excursion, it is convenient to prepend and append extra steps.
The length of the excursion is seen to be twice the total population of the branching process. With this knowledge in hand, it is possible to express the probability of a branching process of a certain size in terms of the probability of a random walk excursion of a certain duration. This example is a special case of what is sometimes known as Dwass’ theorem [2,3].
Mappings such as these often give rise to coincidences in distribution for observables in seemingly unrelated systems. For example, one can condition the above branching process to die out at some finite generation and calculate the distribution of the total population. For a critical branching process, this distribution turns out to be the Kolmogorov-Smirnov distribution. This is the distribution for the Kolmogorov-Smirnov test statistic [4,5], the roughness of an Edwards-Wilkinson interface , the hitting time of a 3d Bessel process , and the size of percolating clusters in the Bethe lattice [8,9].
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Point vortices are idealised sources of vorticity in a 2D Eulerian fluid . Unlike a typical mechanical system in which the Hamiltonian is a function of position and momentum, the point vortex Hamiltonian is a function of position and position. Thus, the spatial configuration of point vortices is, at the same time, the system’s occupation in phase space. Depending on boundary conditions, the motion of a small number of vortices is exactly solvable. For a large number of vortices, a statistical mechanical treatment is appropriate. In the free plane, the generic behaviour of (same sign) vortices is a confined rotation around a centre of “mass”. Examples of configurations for low (left) and high (right) energies are shown below:
A mean field theory can explain some of the qualitative features of these configurations [2,3]. This includes the existence of a negative “temperature”, which reflects the fact that the vortices crowd closer together with increasing energy and therefore occupy less of the phase space. But one clearly observes from simulations that mean field predictions (of e.g. radial occupation densities) deteriorate with increasing energy: vortices can bind to form dipoles, for example. It would be interesting to study the creation and annihilation of these dipoles further. Other questions include the motion of passive tracers that are advected by the velocity field of the vortices. From a far distance, a tracer particle effectively sees a single strong vortex from a collection of vortices, and approximately performs a circular orbit around a centre of rotation. But in and amongst the vortices, the motion of the tracer is far more complicated and might sometimes be described by anomalous diffusion .
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