Arctic circles, random domino tilings and square Young tableaux
Dan Romik
ABSTRACT: It is well-known that domino tilings and certain
other random combinatorial or \break statistical-physics models
on a two-dimensional lattice exhibit a spatial phase transition
between an "arctic" or "frozen" region where the behavior of
the random object is asymptotically deterministic and a
"temperate" region where truly random behavior is observed. One
famous example of such a phenomenon is the Arctic Circle
Theorem due to Jockusch, Propp and Shor, which shows that for
uniformly random domino tilings of a particular region known as
the Aztec Diamond, the curve that forms the interface between
the frozen and temperate regions converges to a circle. Cohn,
Elkies and Propp later derived a more detailed result about the
limiting height function of the typical domino tiling of the
Aztec diamond. In this talk, I will present a new proof of the
Cohn-Elkies-Propp limit shape result for the height function
based on a connection to alternating-sign matrices and a
variational analysis. The proof highlights a surprising
connection of this result to another arctic-circle type
phenomenon observed in a different problem involving uniformly
random square Young tableaux.
BIO: Dan Romik, currently visiting the MSR Theory Group,
obtained his Ph.D. in 2001 from Tel Aviv University and
subsequently did post-doctoral work at Universite Pierre et
Marie Curie, the Weizmann Institute of Science, the
Mathematical Sciences Research Institute (MSRI) and UC
Berkeley. He then held positions at Bell Labs and the Hebrew
University before joining the UC Davis mathematics department
in 2009. His research work is primarily in probability theory
and combinatorics, with a focus on limit shapes of random
combinatorial objects, enumeration, and exactly solvable
lattice models.