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Scaling Limits

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Random motion of a single particle or individual governed by simple stochastic rules is well understood by classical probability theory.

The picture starts to be very different if the rules of motion are more complicated or we study a population of individuals that interact.

Examples include random processes with memory, where the random behaviour of the particle is influenced by the particle's own history, or interacting particle systems, where many simple walkers interact with each other, and various models arising in population biology and ecology where individuals interact with each other as well as possibly the external environment.

Many real life systems can be studied using scaling limits, e.g. crystal growth, spread of infectious diseases, traffic jams, forest fires, spread of advantageous mutations. Mathematical theory often reveals unexpected behaviour.

It is often difficult if not impossible to do calculations for these models, but scaling the parameters may be helpful in identifying limiting behaviour of these systems, about which we may be able to say something interesting.

These limits can be in space or time, or both. Often, taking scaling limits connect many fields in mathematics, such as analysis, combinatorics, complex function theory, and partial differential equations, and is challenging and interesting in itself.

To be able to quantify the phenomenon in some way, which may not be possible otherwise, is an added bonus.