177 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
Quantum Crystals and Spin Chains
In this note, we discuss the quantum version of the melting crystal corner in
one, two, and three dimensions, generalizing the treatment for the quantum
dimer model. Using a mapping to spin chains we find that the two--dimensional
case (growth of random partitions) is integrable and leads directly to the
Hamiltonian of the Heisenberg XXZ ferromagnet. The three--dimensional case of
the melting crystal corner is described in terms of a system of coupled XXZ
spin chains. We give a conjecture for its mass gap and analyze the system
numerically.Comment: 34 pages, 26 picture
Number of Matchings of Low Order in (4,6)-Fullerene Graphs
We obtain the formulae for the numbers of 4-matchings and 5-matchings in
terms of the number of hexagonal faces in (4, 6)-fullerene graphs by studying
structural classification of 6-cycles and some local structural properties,
which correct the corresponding wrong results published. Furthermore, we obtain
a formula for the number of 6-matchings in tubular (4, 6)-fullerenes in terms
of the number of hexagonal faces, and a formula for the number of 6-matchings
in the other (4,6)-fullerenes in terms of the numbers of hexagonal faces and
dual-squares.Comment: This article was already published in 2017 in MATCH Commun. Math.
Comput. Chem. We are uploading it to arXiv for readers' convenienc
Stimulated Raman adiabatic passage-like protocols for amplitude transfer generalize to many bipartite graphs
Adiabatic passage techniques, used to drive a system from one quantum state
into another, find widespread application in physics and chemistry. We focus on
techniques to spatially transport a quantum amplitude over a strongly coupled
system, such as STImulated Raman Adiabatic Passage (STIRAP) and Coherent
Tunnelling by Adiabatic Passage (CTAP). Previous results were shown to work on
certain graphs, such as linear chains, square and triangular lattices, and
branched chains. We prove that similar protocols work much more generally, in a
large class of (semi-)bipartite graphs. In particular, under random couplings,
adiabatic transfer is possible on graphs that admit a perfect matching both
when the sender is removed and when the receiver is removed. Many of the
favorable stability properties of STIRAP/CTAP are inherited, and our results
readily apply to transfer between multiple potential senders and receivers. We
numerically test transfer between the leaves of a tree, and find surprisingly
accurate transfer, especially when straddling is used. Our results may find
applications in short-distance communication between multiple quantum
computers, and open up a new question in graph theory about the spectral gap
around the value 0.Comment: 17 pages, 3 figures. v2 is made more mathematical and precise than v
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