273 research outputs found
How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?
TThe prototypical problem we study here is the following. Given a square, there are approximately ways to tile it with
dominos, i.e. with horizontal or vertical rectangles, where
is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A
conceptually simple (even if computationally not the most efficient) way of
sampling uniformly one among so many tilings is to introduce a Markov Chain
algorithm (Glauber dynamics) where, with rate , two adjacent horizontal
dominos are flipped to vertical dominos, or vice-versa. The unique invariant
measure is the uniform one and a classical question [Wilson
2004,Luby-Randall-Sinclair 2001] is to estimate the time it takes to
approach equilibrium (i.e. the running time of the algorithm). In
[Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven:
for some finite . Here, we go much beyond and show that . Our result applies to rather general domain
shapes (not just the square), provided that the typical height
function associated to the tiling is macroscopically planar in the large
limit, under the uniform measure (this is the case for instance for the
Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our
method extends to some other types of tilings of the plane, for instance the
tilings associated to dimer coverings of the hexagon or square-hexagon
lattices.Comment: to appear on PTRF; 42 pages, 9 figures; v2: typos corrected,
references adde
M2-brane Probe Dynamics and Toric Duality
We study the dynamics of a single M2-brane probing toric Calabi-Yau four-fold
singularity in the context of the recently proposed M-theory crystal model of
AdS4/CFT3 dual pairs. We obtain an effective abelian gauge theory in which the
charges of the matter fields are given by the intersection numbers between
loops and faces in the crystal. We argue that the probe theory captures certain
aspects of the CFT3 even though the true M2-brane CFT is unlikely to be a usual
gauge theory. In particular, the moduli space of vacua of the gauge theory
coincides precisely with the Calabi-Yau singularity. Toric duality, partial
resolution, and a possibility of new RG flows are also discussed.Comment: 50 pages, 24 figures; v2. title changed, abstract and introduction
clarified. to appear in Nucl.Phys.
The symplectic arc algebra is formal
We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one Hochschild cohomology class on a Floer A-algebra associated to the (,)-nilpotent slice obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification . The space is obtained as the Hilbert scheme of a partial compactification of the A-Milnor fiber. A sequel to this paper will prove formality of the symplectic cup and cap bimodules and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.This is the author accepted manuscript. The final version is available from Duke University Press via http://dx.doi.org/10.1215/00127094-344945
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