Two-Color Babylon
We solve the game of Babylon when played with chips of two colors, giving a winning strategy for the second player in all previously unsolved cases.
math.CO↗
arXiv subjects
Peter Winkler
Publications and source records attributed to Peter Winkler.
New Bounds for Edge-Cover by Random Walk
We show that the expected time for a random walk on a (multi-)graph $G$ to traverse all $m$ edges of $G$, and return to its starting point, is at most $2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion on a finite or infinite network of total edge-length $m$.
math.CO↗
Sorting by Placement and Shift
In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of this algorithm is commonly used to hand-sort files.) However, it is not obvious that this algorithm necessarily terminates. We show that in fact the algorithm terminates after at most $2^{n-1}-1$ steps in the worst case (confirming a conjecture of L. Larson), and that there are super-exponentially many permutations for which this exact bound can be achieved. The proof involves a curious symmetrical binary representation.
math.CO↗
Branched Polymers
Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order $n$ in the plane and in 3-space. Our development reveals some more general identities, and allows exact random sampling. In particular we show that a random 3-dimensional branched polymer of order $n$ has diameter of order $\sqrt{n}$.
math.PR↗
Maximum overhang
How far can a stack of $n$ identical blocks be made to hang over the edge of a table? The question dates back to at least the middle of the 19th century and the answer to it was widely believed to be of order $\log n$. Recently, Paterson and Zwick constructed $n$-block stacks with overhangs of order $n^{1/3}$, exponentially better than previously thought possible. We show here that order $n^{1/3}$ is indeed best possible, resolving the long-standing overhang problem up to a constant factor.
math.HO↗
A Solidification Phenomenon in Random Packings
We prove that uniformly random packings of copies of a certain simply-connected figure in the plane exhibit global connectedness at all sufficiently high densities, but not at low densities.
cond-mat.stat-mech↗
Fluid/solid transition in a hard-core system
We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition.
cond-mat.stat-mech↗
Note on Counting Eulerian Circuits
We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P.
cs.CC↗
Hard constraints and the bethe lattice: adventures at the interface of combinatorics and statistical physics
Statistical physics models with hard constraints, such as the discrete hard-core gas model (random independent sets in a graph), are inherently combinatorial and present the discrete mathematician with a relatively comfortable setting for the study of phase transition. In this paper we survey recent work (concentrating on joint work of the authors) in which hard-constraint systems are modeled by the space $\hom(G,H)$ of homomorphisms from an infinite graph $G$ to a fixed finite constraint graph $H$. These spaces become sufficiently tractable when $G$ is a regular tree (often called a Cayley tree or Bethe lattice) to permit characterization of the constraint graphs $H$ which admit multiple invariant Gibbs measures. Applications to a physics problem (multiple critical points for symmetry-breaking) and a combinatorics problem (random coloring), as well as some new combinatorial notions, will be presented.
math.CO↗