5
$\begingroup$

I'm trying to write a program (with javascript or python) that samples a random lozenge tiling of a hexagon with Propp - Wilson's coupling from the past algorithm. I'm quite clear of the framework of the algorithm, but I don't know which is the most efficient way to encode a tiling, or whether different encoding methods would result in non-uniform samplers.

I have at least 3 possible way to encode a lozenge tiling of a hexagon:

  1. View it as a plane partition, i.e. piling cubes in a 3D room. This is quite intuitive, but not easy to implement the add - remove step.

  2. View it as a non-intersecting path system. Each path is further represented by a 0-1 array.

  3. encode it as a interlacing array (which I have not understand it yet). But I have seen at least 3 people that mentioned this approach. For example here

So my question is: which is the best way to encode a tiling, or can anyone explain the 3rd approach?

$\endgroup$
1

1 Answer 1

3
$\begingroup$

You should represent it as a plane partition. But you shouldn't represent a plane partition as a 3D array of 0/1 for absent/present. Most people think of a partition as a set of natural numbers, not a Ferrers diagram. The Ferrers diagram has more symmetry, but usually the set (or decreasing sequence) of numbers is the better choice. Similarly, Wikipedia defines a plane partition as a 2D array of natural numbers, subject to constraints that correspond to the Ferrers diagram not falling over. This is well suited to a computer: it is a compact representation; the compatibilities are inequalities between adjacent entries; adding a box corresponds to incrementing a number.

CFTP requires that you have a Gibbs sampling Markov chain. The way Gibbs sampling works is that you erase part of the diagram, compute all possible ways of filling it in and uniformly choose from this small set. You could probably create a Gibbs sampling Markov chain from any of your representations whose stationary distribution would be uniform.

But CFTP uses more structure, specifically a lattice structure, a partial order such that sup and inf exist. Plane partitions have this structure by product partial order — one partition is bigger than another if all of its components are bigger. If you want to use a different representation, you must check that it has this structure.

Actually, there is a slightly different representation that might be better and has been popular in the past. In the case of regular partitions, it turns the partition 45 degrees, so that the gravity pulling boxes into the corner is now the $y$ axis. The $x$-axis parameterizes diagonal stacks of boxes, touching corner to corner. We record an array index by $x$ of the heights of these stacks. This restores the symmetry of the axes. Similarly, we can represent a (boxed) plane partition as a hexagonal grid of heights, where height is $x+y+z$.

$\endgroup$
4
  • $\begingroup$ There is really only one partial order. But there are many Gibbs chains because every representation gives you a different sense of "local structure." Maybe there is a universal one, where you choose one of the $n^3$ boxes and forget whether it is present. But it is more efficient to group together boxes into $O(n^2)$ sets and forget about all of them at once. In particular, if you use either representation as an array of numbers, you can erase one of those numbers. But for CFTP it is important to make the choice of a replacement number be "the same" for all configurations, compatible w/order. $\endgroup$ Commented Jan 27, 2017 at 15:07
  • $\begingroup$ do you have an example code? if you do, could you share a copy of it? $\endgroup$
    – zemora
    Commented Jan 27, 2017 at 19:34
  • $\begingroup$ I have C code for CFTP for alternating sign matrices. Propp and Wilson have implementations for other cases, including plane partitions. Here are java applets, including plane partitions, but I don't see public source code. It would be good to replace java with js. $\endgroup$ Commented Jan 29, 2017 at 17:32
  • $\begingroup$ thanks. I have emailed Propp to ask for the code but it's rather difficult to understand, it has only a few comments. $\endgroup$
    – zemora
    Commented Jan 30, 2017 at 6:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.