How random and unstable are your phases? (Dec. 7, 2009)
There are phenomena in the natural world that behave randomly or what is seems chaotic such as in percolation and “Brownian movement” of gases. The study of phases in equilibrium among chaotic, random, and unstable physical systems were analyzed first my physicists and then taken on by modern mathematicians. The mathematician Wendelin Werner (Fields Prize) researched how the borders that separate two phases in equilibrium among random, and unstable physical systems behave; he published “Random Planar Curves…”
Initially, the behavior of identical elements (particles) in large number might produce deterministic or random results in various cases. For example, if we toss a coin many times we might guess that heads and tails will be equal in number of occurrences; the trick is that we cannot say that either head or tail is in majority. The probabilistic situations inspire the development of purely mathematical tools. The curves between the phases in equilibrium appear to be random but have several characteristics: first, the curves have auto-similarity, which means that the study of a small proportion could lead to generalization in the macro-level with the same properties of “fractal curves”, the second characteristic is that even if the general behavior is chaotic a few properties remain the same (mainly, the random curves have the same “fractal dimension” or irregular shape; the third is that these systems are very unstable (unlike the games of head and tails) in the sense that changing the behavior of a small proportion leads to large changes by propagation on a big scale. Thus, these systems are classified mathematically as belonging to infinite complexity theories.
Themes of unstable and random systems were first studied by physicists and a few of them received Nobel Prizes such as Kenneth Wilson in 1982. The research demonstrated that such systems are “invariant” by transformations (they used the term re-normalization) that permit passages from one scale to a superior scale. A concrete example is percolation. Let us take a net resembling beehives where each cavity (alveolus) is colored black or red using the head and tail flipping technique of an unbiased coin. Then, we study how these cells are connected randomly on a plane surface. The Russian Stas Smirnov demonstrated that the borders exhibit “conforming invariance”, a concept developed by Bernhard Riemann in the 19th century using complex numbers. “Conforming invariance” means that it is always possible to warp a rubber disk that is covered with thin criss-cross patterns so that lines that intersect at right angle before the deformation can intersect at right angle after the deformation. The set of transformations that preserves angles is large and can be written in series of whole numbers or a kind of polynomials with infinite degrees. The transformations in the percolation problem conserve the proportion of distances or similitude.
The late Oded Schramm had this idea: suppose two countries share a disk; one country control the left border and the other the right border; suppose that the common border crosses the disk. If we investigate a portion of the common border then we want to forecast the behavior of the next portion. This task requires iterations of random conforming transformations using computation of fractal dimension of the interface. We learn that random behavior on the micro-level exhibits the same behavior on the macro-level; thus, resolving these problems require algebraic and analytical tools.
The other case is the “Brownian movement” that consists of trajectories where one displacement is independent of the previous displacement (stochastic behavior). The interfaces of the “Brownian movement” are different in nature from percolation systems. Usually, mathematicians associate a probability “critical exponent or interaction exponent” so that two movements will never meet, at least for a long time. Two physicists, Duplantier and Kyung-Hoon Kwan, extended the idea that these critical exponents belong to a table of numbers of algebraic origin. Mathematical demonstrations of the “conjecture” or hypothesis of Benoit Mandelbrot on fractal dimension used the percolation interface system.
Werner said: “With the collaboration of Greg Lawler we progressively comprehended the relationship between the interfaces of percolation and the borders of the Brownian movement. Strong with Schramm theory we knew that our theory is going to work and to prove the conjecture related to Brownian movement.”
Werner went on: “It is unfortunate that the specialized medias failed to mention the great technical feat of Grigori Perelman in demonstrating Poincare conjecture. His proof was not your tread of mill deductive processes with progressive purging and generalization; it was an analytic and human proof where hands get dirty in order to control a bundle of possible singularities. These kinds of demonstrations require good knowledge of underlying phenomena”. As to what he consider a difficult problem Werner said: “I take a pattern and then count the paths of length “n” so that they do not intersect twice at a particular junction. This number increases exponentially with the number n; we think there is a corrective term of the type n at exponential 11/32. We can now guess the reason for that term but we cannot demonstrate it so far.”
The capacity of predicting behavior of a phenomenon by studying a portion of it then, once an invariant is recognized, most probably a theory can find counterparts in the real world; for example, virtual images techniques use invariance among objects. It has been proven that vision is an operation of the brain adapting geometric invariance that are characteristics of the image we see. Consequently, stability in the repeated signals generates perception of reality. In math, it is called “covariance laws” when system of references are changed. For example, the Galileo transformations in classical mechanics and Poincare transformations in restricted relativity. In a sense, math is codifying the processes of sensing by the the brain using symbolic languages and formulations.
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