Antichains in the random 3-dimensional partial
order
Edward Crane
University of Bristol
Prospects for causal set quantum gravity,
ICMS Edinburgh, September 2015
Antichains in the random 3-dimensional partial order
Acknowledgements
This talk is about a joint project with Nic Georgiou. We got stuck
in 2012, but perhaps with help from someone here we could make
progress.
We had valuable input from Cedric Boutillier, Graham Brightwell,
Rick Kenyon, Malwina Luczak, Mark Walters and Peter Winkler.
Antichains in the random 3-dimensional partial order
Overview
This talk is about a particular model of a random partially ordered
set, called the random 3-dimensional partial order, with a focus on
the rare event that it is an antichain.
This can be viewed as a random interface model, and our
simulations bear a striking similarity to electron scanning
micrographs of surfaces of certain crystalline materials.
In fact, our initial motivation was not solid state physics, but a
question of how to obtain Minkowski space-time as a continuum
limit of random partial orders defined by certain sequential growth
models.
Antichains in the random 3-dimensional partial order
Outline
I
Background on partial orders. Connections to sorting
algorithms.
I
Some models of random partial orders.
I
The random k-dimensional partial order and its properties.
I
Simulation - MCMC and coupling from the past.
I
An approach to understanding the limiting behaviour.
Antichains in the random 3-dimensional partial order
A refresher on partial orders
What is a partial order?
A partial order on a set P is a transitive irreflexive relation ≺ on P.
That is,
(x ≺ y ) ∧ (y ≺ z)
=⇒
x ≺z
and
x ≺y
=⇒
y 6≺ x .
We write x y to mean (x ≺ y ) ∨ (x = y ).
When x 6= y , x 6≺ y and y 6≺ x, we say x and y are incomparable.
Partially ordered sets are sometimes called posets.
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Chains and antichains
A chain in (P, ≺) is a subset C ⊆ P such that every two elements
of C are comparable. That is, the restriction of ≺ to C is a linear
order, or (total order).
The height of (P, ≺) is
sup {|C | : C is a chain in (P, ≺)} .
An antichain is a subset A ⊆ P such that every two elements of A
are incomparable.
The width of (P, ≺) is
sup {|A| : A is an antichain in (P, ≺)} .
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Chain coverings
Suppose P is finite and (P, ≺) is the union of chains C1 , . . . , Ck .
Any antichain can meet each chain in at most one element, so
width(P) ≤ k .
Dilworth’s theorem says this bound is sharp: the minimum
cardinality of a chain covering equals the width of (P, ≺).
Corollary:
|P| ≤ height(P) . width(P)
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Linear extensions
A linear extension of (P, ≺) is a linear order < on P such that
x ≺y
=⇒
x <y.
We are often interested in how many linear extensions (P, ≺) has.
We denote this number by e(P, ≺).
If |P| = n then
1 ≤ e(P, ≺) ≤ n!
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Example: sorting
Suppose we are part of the way through a sorting algorithm to sort
a set P into an unknown linear order, with uniform prior, by
comparing pairs of elements sequentially. The information
gathered so far determines a partial order ≺ on P.
It is natural to consider the set of linear extensions of (P, ≺) as a
probability space with the counting measure, which is the Bayesian
posterior.
Shepp proved that the events x < y and x < z are non-negatively
correlated.
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Greedy sorting
A greedy sorting algorithm might try to choose incomparable
elements x and y as the next two elements to compare so as to
minimise
max (P(x < y ) , P(x > y )) .
By what factor can we guarantee to reduce the number of linear
extensions with each comparison?
Antichains in the random 3-dimensional partial order
A refresher on partial orders
The
2
,
3 3 conjecture
1
When sorting three elements, after one comparison there are three
possible linear extensions. The second comparison either finds the
linear order or leaves two possibilities. This is conjectured to be as
unbalanced as it can be.
Kahn and Kim showed that for a finite partial order there is always
a pair of elements such that P(x < y ) ∈ [3/11, 8/11].
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Intersections of partial orders
The intersection of partial orders <j : j ∈ J on the set P is the
partial order ≺ on P defined by
x ≺y
⇐⇒
(∀j ∈ J) (x <j y ) .
The name makes sense when you think of partial orders as subsets
of P × P.
Every finite partial order is the intersection of its linear extensions.
Antichains in the random 3-dimensional partial order
A refresher on partial orders
The co-ordinatewise partial order
The co-ordinatewise partial order on Rk is the intersection of the
linear orders on the individual co-ordinates. So
(x1 , . . . , xk ) ≺ (y1 , . . . , yk )
⇐⇒
xi < yi , i = 1, . . . , k .
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Dimension
The dimension of a partial order (P, ≺) is the least k for which it
is isomorphic to the induced partial order on a subset of the
co-ordinatewise partial order on Rk .
Equivalently, the dimension of (P, ≺) is k if ≺ is the intersection
of k linear orders on P but not of k − 1 linear orders.
Antichains in the random 3-dimensional partial order
A refresher on partial orders
Up-sets and down-sets
The up-set of a subset X ⊂ P is
Up(X ) = {p ∈ P : (∃x ∈ X )(x p)}
and the down-set is
Down(X ) = {p ∈ P : (∃x ∈ X )(p x)} .
Antichains in the random 3-dimensional partial order
Random partial orders
Some random partial orders
Several different models of random finite partial orders have been
studied. For example:
I
Take a partial order uniformly from all partial orders on [n].
Kleitman and Rothschild (1975) showed that asymptotically
almost surely the uniform random partial order has height 3.
I
Take a random graph G on the vertex set [n] and take the
smallest partial order satisfying i ≺ j whenever i < j and
(i, j) ∈ E (G ). When G = G (n, p) this is called transitive
percolation.
Antichains in the random 3-dimensional partial order
Random partial orders
Sequential growth processes
I
(Rideout and Sorkin). Start with the empty partial order and
repeatedly add a new maximal element, choosing the
antichain of elements that it covers in some random way. The
result is a random sequence of partial orders ([n], ≺n ), each
one inducing its predecessors.
I
Particularly interesting are order-invariant growth processes,
where for each n and each isomorphism class of partial orders
on n elements, each possible labelling of that partial order by
a linear extension is equally likely to occur.
Antichains in the random 3-dimensional partial order
Random partial orders
A question
Is there an order-invariant growth process which produces the same
random infinite partial order as a unit-intensity Poisson process
(sprinkling) in the positive light-cone of the origin in
(1 + 1)-dimensional Minkowski spacetime?
What are the Gibbs measures on linear extensions of this
sprinkling? Is there one Gibbs measure for each Lorentz frame?
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The random k-dimensional partial order
The random k-dimensional partial order Pk (n) is the intersection
of k independent uniform random linear orders on [n].
Equivalently it is the co-ordinatewise partial order on a set of n
i.i.d. points x1 , . . . , xn ∈ [0, 1]k , distributed according to Lebesgue
measure.
We will consider the limit where n → ∞ with k fixed.
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The random k-dimensional partial order
The random k-dimensional partial order Pk (n) is the intersection
of k independent uniform random linear orders on [n].
Equivalently it is the co-ordinatewise partial order on a set of n
i.i.d. points x1 , . . . , xn ∈ [0, 1]k , distributed according to Lebesgue
measure.
We will consider the limit where n → ∞ with k fixed.
The dimension of Pk (n) is k, asymptotically almost surely (a.a.s.)
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The height Hk (n) of Pk (n)
P (Hk (n) ≥ m) is bounded by the expected number of m-element
chains, which is
n
n
P (Hk (m) = m) =
(m!)1−k .
m
m
This is o(1) when m = en1/k , and decays exponentially in λ if we
take m = λn1/k .
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The height Hk (n) of Pk (n)
Divide [0, 1]k into mk subcubes, where m ≈ n1/k .
Consider the m subcubes along the major diagonal of the cube.
With probability close to 1 at least half of these contain a point of
Pk (n), so that
Hk (n) ≥ m/2 .
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The height Hk (n) of Pk (n)
In fact, Hk (n) is sharply concentrated about its mean value:
Theorem (Bollobás and Brightwell, 1992)
For each integer k ≥ 2, there are constants ck , Ck such that for n
sufficiently large,
!
1/2k log3/2 n
λC
n
2
k
P Hk (n) − ck n1/k >
≤ 80λ2 e −λ
log log n
for every λ with 2 < λ < n1/2k / log log n. Here c2 = 2 and each
ck ≤ e.
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The width Wk (n) of Pk (n)
Recall that by Dilworth’s theorem,
n ≤ Wk (n) . Hk (n) .
Therefore
Wk (n) ≥
n
e n1/k
a.a.s.
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
Wk (n) is concentrated about its mean
Theorem (Brightwell, 1992)
There is a constant C such that, for each fixed k, with probability
tending to 1 as n → ∞ we have
7
1√
k − C n1−1/k ≤ Wk (n) ≤ kn1−1/k .
2
2
Moreover, for each integer k ≥ 2 there is a constant Dk such that,
for n sufficiently large,
!
λDk n1/2−1/2k log n
2
P |Wk (n) − EWk (n)| >
≤ 4λ2 e −λ .
log log n
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The probability that Pk (n) is an antichain
To find m such that P (Wk (n) > m) = o(1), as we did for Hk (n),
we need an upper bound for the probability that a given m-element
subset is an antichain.
Define
Qk (n) = P (Hk (n) = 1) .
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The expected number of linear extensions
Define
Ek (n) = e(Pk (n)) .
Points x1 , . . . , xn ∈ [0, 1]k form an antichain if and only if the order
in the k th coordinate is the reverse of a linear extension of the
coordinatewise partial order on their projections to the first k − 1
coordinates.
Therefore for k ≥ 2 we have
Qk (n) =
Ek−1 (n)
n!
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The expected number of linear extensions
Theorem (Brightwell, 1992)
Qk (n) ≤
21/k k (k+1)/(k−1) n−1/(k−1)
and hence
EEk (n) ≤
2kn1−1/k
n
.
n
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
Brightwell’s towers method
The proof uses a subdivision of [0, 1]k into mk subcubes. The
subcubes are given a partial order so that
x1 ∈ C1 , x2 ∈ C2 , C1 ≺ C2 =⇒ x1 ≺ x2 .
The subcube order has a chain covering by t = mk − (m − 1)k
‘diagonal towers’. Every maximal antichain of subcubes meets
each tower. There are at most 2t maximal subcube–antichains,
each of volume t/mk . Therefore
n
Qk (n) ≤ 2t . t/mk
Brightwell’s bound is obtained by optimising this bound over m.
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
log Ek (n) is concentrated about its mean
Theorem (Brightwell, 1992)
n
k −1
log n − 2
k
≤ E log Ek (n) ≤ n
k −1
log n + log(2k) .
k
Moreover, for any k, n, and any real λ,
P |log Ek (n) − E log Ek (n)| > λn1/2 log n ≤ 2 exp −λ2 /2 .
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The 2-dimensional random partial order
A sample of P2 (n) is described by a uniform random element
σ ∈ Sn , giving the order of the 2nd coordinate when the points are
ordered by their first coordinate.
The height is the length of the longest increasing subsequence in
σ. The width is the length of the longest decreasing subsequence
in σ. The distribution of these is the subject of Ulam’s problem.
Logan and Shepp (1977) and Vershik and Kerov (1977) showed
that
√
E (H2 (n)) ∼ 2 n .
Baik, Deift, and Johansson (2000) determined the scaling limit of
the distribution of H2 (n). It is a Tracy-Widom distribution on the
scale n1/6 .
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
Linear extensions of 2-dimensional partial orders
Theorem (Bollobás, Brightwell and Sidorenko)
With probability tending to 1 as n → ∞,
r
n
π 1/2
1 1/2 n
n
.
(1 − o(1)) √ n
≤ E2 (n) ≤ (1 + o(1))
2e
e
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
Linear extensions and complements
The proof of the lower bound relies on a beautiful inequality of
Sidorenko.
Let P be a complement of P, both on the set [n]. This means that
x and y are comparable in P exactly when they are incomparable
in P. Then
e(P)e P ≥ n!
Equality holds exactly when P is a series-parallel partial order.
In fact e(P) depends only on the comparability graph of P, since it
is the volume of the chain polytope of P.
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
Linear extensions of 2-dimensional partial orders
Let P be the random partial order given by the intersection of two
permutations (1, . . . , n) and a uniform random permutation π. Let
P be the intersection of (n, . . . , 1) and π. Then P is a complement
of P and both P and P have the law of P2 (n).
E (E2 (n)) = E
e(P) + e P
2
!
≥ E e(P) . e P
1/2
≥ (n!)1/2
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
The probability that P3 (n) is an antichain
The bounds described above give
n
n
(1 − o(1))e 1/2 n−1/2
≤ Q3 (n) ≤ 21/3 32 n−1/2 .
The constants here are e 1/2 ≈ 1.65 and 21/3 32 ≈ 11.34.
With a more careful analysis using the dimer model we improved
the constant in the upper bound to
p
3 3e(3 log 3 − 4 log 2) ≈ 6.2,
and then (with Cédric Boutillier) to 3.443.
Does limn→∞ Q3 (n)1/n exist? If so can we compute it?
Antichains in the random 3-dimensional partial order
The random k-dimensional partial order
Typical antichains
Condition on the event that the first n − 1 points form an
antichain to see that Q3 (n)/Q3 (n − 1) is the expected volume of
[0, 1]3 \ (Down(X ) ∪ Up(X )) ,
where X has the law of P3 (n − 1) conditioned to be an antichain.
Therefore to understand the asymptotics of Q3 (n) we would like to
understand the shape of a typical antichain.
Antichains in the random 3-dimensional partial order
Simulation
An MCMC simulation
We simulated P3 (n) conditioned to be an antichain using a
rejectionless MCMC method.
The algorithm takes a random walk around a graph on the set of
pairs of permutations (π1 , π2 ) such that [n] ∩ π1 ∩ π2 is an
antichain. This chain is irreducible and aperiodic, and the invariant
measure assigns mass to an antichain proportional to the number
of available moves.
We have not yet proved a bound on the mixing time of the chain,
so we don’t know how close our sampling was to uniform.
Antichains in the random 3-dimensional partial order
Simulation
Simulation of P3 (1000) conditional on being an antichain
Antichains in the random 3-dimensional partial order
Simulation
Simulation of P3 (10000) conditional on being an antichain
Antichains in the random 3-dimensional partial order
Simulation
Perfect uniform sampling of linear extensions
Karzanov and Khachiyan (1991) gave a MCMC method for
sampling from the linear extensions of a partial order on n
elements, with mixing time O(n5 log n). Bubley and Dyer (1999)
improved this to O(n3 log n) with a slightly different chain.
Algorithms for exact uniform sampling of linear extensions of a
finite partial order:
I
I
Felsner and Wernisch (1997): 2-dimensional partial orders
Huber (2006): general partial orders
Both algorithms use coupling from the past, so they have random
running time, with geometric tail. The expected running time is
polynomial in n.
Unforunately we have not been able to find a grand coupling for a
Markov chain on the space of n-point antichains in [0, 1]3 .
Antichains in the random 3-dimensional partial order
Simulation
Maximum antichains in P3 (n)
Theorem
The set of maximum-sized antichains in a finite partial order P
forms a distributive lattice, and therefore can be sampled perfectly
using the CFTP method of Propp and Wilson.
Our perfect simulations of uniformly-sampled
maximum-size
antichains from P3 (n), of size Θ n2/3 , are indistinguishable to the
eye from our simulations of P3 (n) conditioned to be an antichain.
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
A probabilistic lower bound for Q3 (n)
Suppose E ⊂ [0, 1]3 is a measurable set with Vol(E ) > 0. Let XE
be a set of n i.i.d. points uniformly distributed in E .
P(H3 (n) = 1) ≥ Vol(E )n . P (XE is an antichain)
For a given n and > 0, which set E should we choose with
Vol(E ) = in order to maximise this lower bound?
Defining E by x1 + x2 + x3 ∈ 23 − δ, 32 + δ leads to the bound
Q3 (n)1/n ≥
9
√
4e 2n
.
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
A two-point question
For a fixed volume , which set E minimises P(x ≺ y ) for two
independent points x and y uniformly distributed in E ?
When E is a tubular neighbourhood of a fixed surface S defined by
|x3 − f (x1 , x2 )| < h(x1 , x2 ) ,
R
where f and h are smooth functions and h dx1 dx2 = 1, we have
Z
P(x ≺ y ) ≈ c.
h3
dx1 dx2 .
f1 f2
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
A two-point question
A variational argument shows that the optimal h for a given
smooth f is
|f1 f2 |1/3
h= R
|f1 f2 |1/3 dx1 dx2
which means that we want to maximise
Z
|f1 f2 |1/3 dx1 dx2 .
over smooth f satisfying appropriate boundary conditions.
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
A two-point question
This has a unique extremum, which is the solution of the PDE
fxy
fyy
fxx
−
+ 2 = 0
2
fx
fx fy
fy
with f (0, y ) = 1, f (1, y ) = 0, f (x, 0) = 1 and f (x, 1) = 0.
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
The limit surface
In fact we believe the surface x3 = f (x1 , x2 ) is the limit surface of
P3 (n) conditioned to be an antichain, in the sense that there is a
probability measure µ supported on the surface which is the limit
in probability of the empirical distributions of P3 (n) conditioned to
be an antichain, as n → ∞.
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
An approach to proving a large deviations principle
The space of probability measures on [0, 1]3 is compact in the
weak topology. We cover it with open sets.
One of these is a small ball U0 about µ in the Prohorov metric.
Each other set U in the cover does not contain µ; we have to show
that the probability that the empirical measure of P3 (n)
conditioned to be an antichain lies in U tends to 0 asymptotically
faster than for U0 .
Antichains in the random 3-dimensional partial order
The limit surface of P3 (n) conditioned to be an antichain
An approach to proving a large deviations principle
It is not too hard to reduce to the case where U is a small
neighbourhood of a probability measure ν that is supported on an
antichain and has uniform one-dimensional marginals.
The idea is to cover the support of ν by finitely many thin slabs.
We then have to bound the probability that nearly all of our n
points are appropriately distributed among the slabs, and the
restriction to each slab is an antichain.
The aim is to choose the slabs so that this is essentially the same
problem for each slab.