Third meeting, Technion, 29.4.19, Amado Building room 232
Rigidity of proper colorings of Z^d (and other graph homomorphisms)
A proper q-coloring of Z^d is an assignment of one of q values to each vertex of Z^d such that adjacent vertices are assigned different values. Such colorings arise naturally in combinatorics, ergodic theory (as a subshift of finite type) and statistical physics (as the ground states of the antiferromagnetic q-state Potts model). How does a "uniformly picked" proper q-coloring of Z^d look like? To make sense of this question, one may sample uniformly a proper q-coloring of a large finite domain in Z^d and seek possible patterns in the resulting coloring. Alternatively one may seek to classify the translation-invariant probability measures on proper q-colorings having maximal entropy. Work of Dobrushin (1968) implies that when q is large compared with d, q-colorings support a unique measure of maximal entropy. We focus on the opposite regime, when d is large compared with q, and prove that long-range order emerges in typical colorings, leading to multiplicity of measures of maximal entropy. Concretely, the extremal measures of maximal entropy are in correspondence with partitions (A,B) of the q colors into two equal-sized sets (near-equal if q is odd), and in the measure corresponding to (A,B) most of the vertices of the even (odd) sublattice of Z^d are assigned values from A (from B).
The results address questions going back to Berker-Kadanoff (1980), Kotecky (1985) and Salas-Sokal (1997).
The methods extend to the study of other graph homomorphism models on Z^d satisfying a certain symmetry condition and to their "low temperature" versions.
Joint work with Yinon Spinka.
On tiling the real line by translates of a function
If f is a function on the real line, then a system
of translates of f is said to be a tiling if it constitutes
a partition of unity. Which functions can tile the line by
translations, and what can be said about the structure of the
tiling? I will give some background on the problem and present
our results obtained in joint work with Mihail Kolountzakis.
The optimal lifting in SL(3)
Sarnak proved in one of his letters that as q goes to infinity, for every $\epsilon>0$ almost every matrix in $SL_2(F_q)$ can be lifted to a matrix in $SL_2(Z)$, where every coordinate is bounded by q^(3/2+\epsilon). The exponent 3/2 is optimal, since the number of matrices in $SL_2(Z)$ with coordinates bounded by $T$ is asymptotic to $T^2$.
We prove a similar theorem, with optimal exponent, in the context of the action of $SL_3(Z)$ on the projective 2-dimensional space over $F_q$. Our work is based on lattice point counting arguments as in the work of Sarnak and Xue, and on property T for SL3.
We will also explain the relation of this theorem to the work of Ghosh, Gorodnik and Nevo, and to some general conjectures of Sarnak whose aim
is to "approximate" the Generalized Ramanujan Conjecture to deduce Diophantine results.
Based on joint works with Konstantin Golubev and Hagai Lavner.
Graded graphs and central measures
Graded graphs (Bratteli diagrams) as dynamical language : Markov compact of paths, central measures, connections to asymptotic representation theory of locally finite groups, characters as central measures, limit-shape type theorems and generalized law of large numbers.
You are cordially invited.
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