More on Princeton Companion to Mathematics aka ‘PCM’.
Quite a few of the single Companion chapters are available from the authors’ own web sites… I gave links to Terry Tao and Tim Gowers blogs below.
Gordon Slade wrote the chapter ‘Probabalistic Models in Critical Phenomena‘
Its a nice overview of whats been happening in this area, with sections about Critical Phenomena, Branching Processes, Random Graphs, Percolation, Ising Model, Random Cluster model and also introduces SLE the Stochastic-Loewner-Evolution work by Oded Schramm, Wendelin Werner etc.
If you’ve never heard of this stuff heres a motivator of sorts, by yours truly –
You can imagine a hexagonal grid, where there are islands of light grey sand scattered amongst a sea of darker blue hexagons… and the parameter p defines the probability of a hex being dry land. Lets say an islander can jump 3 hex grids but no more… initially the probability of getting to any other island is 0, and as we increase p, islanders can sometimes hop across a few islands… at some point, when p [the ratio of sandy islands to sea blue hexagons] is high enough, islanders will be able to travel from one side of the world to another by hopping – so, at some critical value of p, a whole new behaviour is possible. This sounds like a plausible analogy describing, say, the breakdown of a dielectric in a large electric field, or a metal fracture occurring under pressure.. a lot of physical phenomena could be described by this kind of critical threshold.
Ill go out on a limb here and simply state – these critical phenomena are so central to Nature that they _must_ occur in the financial markets also. So one could imagine, at some point when a certain number of people are selling off a stock, that beyond that point an avalanche will occur where everyone jumps on-board and dumps the stock… its a blunt, unsophisticated example, but it does seem plausible that this new model of Critical Phemomena might help us in Quantitative Finance…
It seems that Terry Taos additive combinatorics should be applicable here too.. basically at some point, the islands become dense enough that we can say by using combinatorics that certain things must occur…
SLE theory looks at conformal maps of a Brownian motion, so there might be results relating back to PDE [like Feynman-Kac] and Options pricing under Wiener diffusions.
Theres a good intro to SLE by Greg Lawler here