Clayton Shonkwiler PRO
Mathematician and artist
Polyhedra, Sampling Algorithms for Random Polygons, and Applications to Ring Polymer Models
/ag17
This talk!
Statistical physics Viewpoint
A ring polymer in solution takes on an ensemble of random shapes, with topology (knot type!) as the unique conserved quantity.
Knotted DNA
Wassermann et al.
Science 229, 171–174
Knot complexity in DNA from P4 tailless mutants
Arsuaga et al., PNAS 99 (2002), 5373–5377
Is this surprising?
Statistical physics viewpoint
A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.
Modern polymer physics is based on the analogy between a polymer chain and a random walk.
– Alexander Grosberg
Protonated P2VP
Roiter/Minko
Clarkson University
Plasmid DNA
Alonso–Sarduy, Dietler Lab
EPF Lausanne
Sampling random walks is easy
Generate \(n\) independent uniform random points on \(S^2\) and treat them as an ordered list of edge vectors.
...but sampling random polygons is hard
Alvarado, Calvo, Millett, J. Stat. Phys. 143 (2011), 102–138
Ansatz
Random Polygon \(\Leftrightarrow\) point in some (nice!) configuration space
Knowledge of the (differential, symplectic, algebraic) geometry of these conformation spaces leads to both
theorems and fast numerical algorithms for studying and comparing polygons in \(\mathbb{R}^3\).
A random closed \(n\)-edge polygon is a \(k\)-edge random walk and an \((n-k)\)-edge random walk, conditioned on having the same end-to-end distance.
Key Idea
Density of the end-to-end distance
Classical Fact: The density of the end-to-end vector of an \(n\)-step random walk is
\phi_n(\ell) = \frac{2\ell}{\pi} \int_0^\infty x \sin \ell x \,\mathrm{sinc}^n x \,\mathrm{d}x
= \frac{2^{-n-1}}{\pi
\ell(n-2)!} \sum_{k=0}^{n-1} (-1)^k \binom{n-1}{k} \left([-2k+\ell +n-2]_+^
{n-2}\right.
\left.-
[-2 k+\ell +n]_+^{n-2}\right)
Proof: Fourier transform (since \(\mathrm{sinc}\) is the transform of the boxcar function).
This is piecewise-polynomial in \(\ell\) of degree \(n-3\)
Density of a polygon chord
Proposition (with Cantarella): The pdf of the chord connecting \(v_1\) with \(v_{k+1}\) in an \(n\)-gon is
\frac{4 \pi \ell^2}{C_n} \phi_k(\ell) \phi_{n-k}(\ell)
where \(C_n = 2^{n-5}\pi^{n-4} \int_{-\infty}^{\infty} x^2 \,\mathrm{sinc}^n x \,\mathrm{d}x\).
Fact: This is piecewise-polynomial in \(\ell\) of degree \(n-4\).
Expected values of chordlengths
Questions
Why are the expectations rational?
Why degree \(n-4\)?
Where the heck are the polyhedra?
Continuous symmetry \(\Rightarrow\) conserved quantity
Some classical mechanics
Duistermaat–Heckman Theorem (stated informally): On a \(2m\)-dimensional symplectic manifold, \(d\) commuting Hamiltonian symmetries (a Hamiltonian \(T^d\)-action) induce \(d\) conserved quantities (momenta).
The joint distribution of the momenta is continuous, piecewise polynomial, degree \(\leq m-d\).
\(n\)-gons up to rotation are \(2m=(2n-6)\)-dimensional, so chord length is piecewise polynomial of degree \(\leq\)
\(m-1=(n-3)-1=n-4\)
We actually have more symmetries!
\(n-3\) commuting symmetries
Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.
Main Theorem
Theorem (with Cantarella, ’16): The joint distribution of \(d_1,\ldots , d_{n-3}\) and \(\theta_1, \ldots , \theta_{n-3}\) are all uniform on their domains.
Proof: Check that D–H applies (this is the hard part, since the torus action is not defined everywhere).
Then count: \(m=n-3\) and we have \(d=n-3\) symmetries, so the pdf of the \(d_i\) is piecewise polynomial of degree \(\leq\)
\(m-d = (n-3)-(n-3)=0\).
Since the pdf is continuous and the domain is connected, it must be constant.
A polytope (finally!)
The \((n-3)\)-dimensional moment polytope \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities
0 \leq d_i \leq 2
1 \leq d_i + d_{i-1}
|d_i - d_{i-1}| \leq 1
0 \leq d_{n-3} \leq 2
A sampling algorithm
Theorem (with Cantarella, Duplantier, Uehara, ’16): A direct sampling algorithm for equilateral \(n\)-gons with expected performance \(O(n^{5/2})\).
If we let \(s_i = d_i - d_{i-1}\) for \(i=1,\ldots , n-2\) and \(s_i \in [-1,1]\), then we have \(|d_i - d_{i-1}|\leq 1\).
Proposition (with Cantarella, Duplantier, Uehara): If we build \(d_i\) from \(s_i\) sampled uniformly from the hypercube \([-1,1]^{n-3}\), the \(d_i\) obey the triangle inequalities with probability asymptotic to
6 \sqrt{\frac{6}{\pi}} n^{-3/2}.
RandomDiagonals[n_] :=
Accumulate[
Join[{1}, RandomVariate[UniformDistribution[{-1, 1}], n]]];
InMomentPolytopeQ[d_] :=
And[Last[d] >= 0, Last[d] <= 2,
And @@ (Total[#] >= 1 & /@ Partition[d, 2, 1])];
DiagonalSample[n_] := Module[{d},
For[d = RandomDiagonals[n], ! InMomentPolytopeQ[d], ,
d = RandomDiagonals[n]];
d[[2 ;;]]
];
Diagonal sampling in 3 lines of code
Not the first direct sampler (Grosberg–Moore, Diao–Ernst–Montemayor–Ziegler), but simple and fast.
An explicit grid on the moment polytope
If we round every point \((d_1),\ldots , d_{n-3})\) in the moment polytope to the nearest point with half-integer coordinates \(\frac{1}{2}(x_1, \ldots , x_{n-3})\) …
Orthoscheme decomposition
… and subdivide the regions of attraction by
ordering and sign of \(d_i - \frac{1}{2} x_i\), we get a collection of identical, orthogonal simplices of equal area (this works for any \(n\)):
There are lots of simplices…
Proposition*: The number of half-integer points in \(\mathcal{P}_n\) is the Motzkin number
M_{n-3} = \sum_{k=0}^{\lfloor \frac{n-2}{2}\rfloor} \binom{n-2}{2k} C_k
\sim \sqrt{3} \frac{1 + \frac{1}{16(n-3)}}{(2n-3)\sqrt{(n-1)\pi}} 3^{n-2}
Therefore, there are exponentially many simplices in this decomposition. ☹️
… clever data structures to the rescue
Theorem (K. Chapman, in progress): With \(O(n^3)\) preprocessing of the transition probabilities, a direct sampling algorithm for equilateral \(n\)-gons in \(O(n^2 (\log n)^2)\) time.
The simplices are indexed by certain permutations.
We can recursively construct Lehmer codes of valid permutations using a Markov chain.
Questions
- What’s the corresponding story for planar \(n\)-gons?
- Polygons form a Kähler manifold, so there’s a well-defined distance between \(n\)-gons. How should we optimally register polygons? Can we find explicit geodesics?
- What about more complicated polymer topologies?
Thank you!
References
The symplectic geometry of closed equilateral random walks in 3-space
J. Cantarella & C. Shonkwiler
Annals of Applied Probability 26 (2016), no. 1, 549–596
A fast direct sampling algorithm for equilateral closed polygons
J. Cantarella, B. Duplantier, C. Shonkwiler, & E. Uehara
Journal of Physics A 49 (2016), no. 27, 275202
Sampling knots using orthoschemes
K. Chapman
In preparation
Polytopes and Polygons
By Clayton Shonkwiler
Polytopes and Polygons
Polyhedra, sampling algorithms for random polygons, and applications to ring polymer models
