Richard Kenyon
On the uniform cycle-rooted spanning tree in Z^2.
Fri, 03/23/2012 - 15:01 — SomNambulAuthors: Richard Kenyon, Wei Wu, Adrien Kassel
Subjects: Probability
AbstractWe compute the asymptotics of the first and second moments of the area of the
cycle of a random cycle-rooted spanning tree (spanning unicycle) of any
sequence of graphs $G_n\subset {\mathbb Z}^2$, such that $\frac{1}{n}G_n$
approximates a bounded domain $D\subset{\mathbb C}$. We show that the first and
second moments grow like $\frac{4}{\pi}\log n$ and $C\cdot\text{Area}(D)n^2$,
respectively, for an explicit constant $C=C(D)$.- Read more
- 2 comments
- login or register to post comments
Hausdorff dimension for fractals invariant under the multiplicative integers.
Mon, 02/28/2011 - 16:01 — SomNambulAuthors: Yuval Peres, Boris Solomyak, Richard Kenyon
Subjects: Dynamical Systems
AbstractWe consider subsets of the (symbolic) sequence space that are invariant under
the action of the semigroup of multiplicative integers. A representative
example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$
for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets
and show that they are typically different. The proof proceeds via a
variational principle for multiplicative subshifts.Spanning forests and the vector bundle laplacian.
Mon, 01/25/2010 - 16:01 — SomNambulAuthors: Richard Kenyon
Subjects: Probability
AbstractThe classical matrix-tree theorem relates the determinant of the
combinatorial laplacian on a graph to the number of spanning trees. We
generalize this result to laplacians on one- and two-dimensional vector
bundles, giving a combinatorial interpretation of their determinants in terms
of so-called cycle rooted spanning forests. We construct natural measures on
CRSFs for which the edges form a determinantal process.Lectures on Dimers.
Mon, 10/19/2009 - 13:48 — SomNambulAuthors: Richard Kenyon
Subjects: Probability
AbstractThese are lecture notes for lectures at the Park City Math Institute, summer
2007. We cover aspects of the dimer model on planar, periodic bipartite graphs,
including local statistics, limit shapes and fluctuations.Lectures on Dimers.
Mon, 10/19/2009 - 13:48 — SomNambulAuthors: Richard Kenyon
Subjects: Probability
AbstractThese are lecture notes for lectures at the Park City Math Institute, summer
2007. We cover aspects of the dimer model on planar, periodic bipartite graphs,
including local statistics, limit shapes and fluctuations.
User login
