Management of Complex Systems
R.S.MacKay@warwick.ac.uk, Mathematics

Determine upper bounds on the range of influence in space and time of local control on a probabilistic cellular automaton in the weakly dependent regime.

Weakly dependent PCA have a unique stationary probability that attracts every initial distribution and exponentially so, e.g. RS MacKay, Robustness of Markov processes on large networks, J Difference Eqns & Applns 17 (2011) 1155–67.  If the dependency matrix satisfies an exponential decay upper bound it looks plausible that the effects of local control on the probability distribution should also decay exponentially in space.  Compare proof for the direct product case in Baesens C, MacKay RS, Exponential localization of linear response in networks with exponentially decaying coupling, Nonlinearity 10 (1997) 931–940.  For PCA we need a tensor product version of this.

Try to derive such a bound.

This would be beneficial to policy makers in many contexts, because an understanding that the effects of local control may not go very far is important to save wasting effort.  Of course, outside the weakly dependent regime things could be very different, but it puts the emphasis on determining which regime a system is in.

This could lead to a PhD tackling the question of the range of control of probability distributions for sample systems outside the weakly dependent regime, say with two phases.  A possible co-supervisor might be Vassili Kolkoltsov (Statistics).