The renormalization fixed point as a mathematical object


Appeared in Twenty years of Bialowieza: A Mahtematical anthology, World Scientific Monograph Series in Mathematics . 8

Author's comments: This paper was prepared for a meeting in Bialowieza that I was unable at the last minute to attend. It has appeared in the proceedings of that conference, Twenty years of Bialowieza: A mathematical anthology. The paper was intended as a beginning. Several years of work, largely numerical and very often in collaboration, on percolation and the Ising model were an attempt on my part to get a handle on what was for me their mathematically fascinating aspect, referred to as renormalization: the observed behavior of large systems for which repeated re-scaling is possible can be described by a very small number of parameters, and the convergence under re-scaling of the values of these parameters to their limits is extremely rapid. I have never found in the literature or discovered on my own any method of any general promise for defining these parameters or for demonstrating their properties. I had hoped when writing this paper to have within my grasp some promising ideas. I thought about them, but either I did not think long enough or hard enough, or they were worth less than I thought. Whatever it was, my attention has been distracted for several years by other matters, every bit as difficult and intractable as renormalization, so that I have not been able to return to it. This was likely no loss to science, but I regret it, for the mathematical questions are in my view of profound interest. I still harbor a little hope that in the coming years I can return to them.