# Interactive simulations

Polya’s urn in this model, starting with one red ball and one black ball in an urn, we repeatedly select a ball at random from the urn and return it, while also adding another ball of the same colour. The probability of selecting a red ball is proportional to the number of red balls raised to the power alpha. Observe the different behaviour as we vary alpha. There is a phase transition when alpha=1.

WARM process on a graph at each step of this process we choose a vertex at random, and from the chosen vertex we choose one of the edges (colours) with probability proportional to the number of times (plus 1) that the edge has previously been selected, raised to the power alpha. Now look at the set of edges that are chosen infinitely often. There are in fact infinitely many phase transitions for this model! The randomise button allows us to change the vertex selecction probabilities from uniform to “random”.

Exclusion process on a circle in this model, particles try to move around a circle, with a given probability of trying to go clockwise. When a particle tries to make a move, it fails if there is already another particle in the spot that it wanted to move to. How fast does each particle move around the circle when there are many particles?

The voter model in this model, each site carries a different vote (colour). Some of the voters may be obstinate (they will never change their vote) – you can set these vertices by clicking on them – while others are influenced by their neighbours. If none of the voters are obstinate (or all obstinate voters are of the same colour), then eventually all voters will agree.