Let's read Danica's college paper

My other hero mathematician, Danica McKellar, co authored this paper:

Perculation and Gibbs states

Abstract. For a region of the nearest-neighbour ferromagnetic Ashkin–Teller spin systems on Z^2 , we characterize the existence of multiple Gibbs states via percolation. In particular, there are multiple Gibbs states if and only if there exists percolation of any of the spin types (i.e. the magnetized states are characterized by percolation of the dominant species). This result was previously known only for the Potts models on Z^2

What's a Gibbs state?

Wiki knows all: In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system. For example, a stationary or steady-state distribution of a Markov chain, such as that achieved by running a Markov chain Monte Carlo iteration for a sufficiently long time, is a Gibbs state.

My hero Danica McKellar, thinks everything has a banker bot maintaining a probability graph so the equipartition is maintained and we can use set which will appear, just not now.  By the way, have we seen William Gibbs recently?

Here is is, the spitting image of Sigmund Freud.  I guess we have to call him one of the inventors of the theory of everything. He has a kind of quantum set theory.

So Danica knows percolation theory, which fundamentally, tells us how many shapes of the probability graph we can have. A perculation is an open path to the top of the tree. Danic plays the Wythoff game.



So, here,

She and Matilde Marcolli should be great friends, one would think. Both are authors and mathematicians who get the toe. They must have met at some book fair somewhere. They both know about the other, I am sure. 

Danica tells us:

We begin with a description of a general Ashkin–Teller model on an arbitrary graph with spins at each vertex. There are four possible spin types, labelled: blue, red C , yellow, and red − . The spins may be regarded as lying equidistant on the unit circle, occurring clockwise in the order just named, with blue at 12 o’clock. There is complete symmetry around the circle, so that interactions receive energy assignments based solely on the relative positions of the spin colours on the circle. Here the model is ‘completely’ ferromagnetic: colours opposite to each other receive the highest energy assignments; the like–like interactions the lowest, and the adjacent colours receive an intermediate energy. Without loss of generality, we may set this intermediate energy level D 0. For positive K h i;j i , k h i;j i , we set the like–like interaction between sites i and j along the edge h i;j iD k h i;j i − K h i;j i , and the interaction for spin pairs with colours opposite to each other to k h i;j i . Although the Z 2 Ashkin–Teller model in our theorem has uniform couplings (and at most one edge between any two sites), some of our proofs will use the flexibility of multiple edges between sites and nonuniform coupling constants. In this paper, we confine attention to the parameter region k h i;j i 6 K h i;j i = 2 for all h i;j i .

Sounds like spinning tops on a circle!  How did they get there?  She proves equipartition, she proves the existence of a quantum set model the guarantees enough queue to cover the the descent of four tops from Gibbs mountain.  The tops can wait in the queue for paths with finite capacity. 

I should help Danica write a book! She is my first or second hero. Is my selection biased?