A thought about pancakes

The pancake sorting problem is to determine the diameter of the pancake graph, the Cayley graph of {S_n} with respect to the generating set {P_n = \{p_1, \dots, p_n\}}, where {p_k} is defined (in two-line notation) by

\displaystyle  p_k = \left(\begin{array}{ccccccc} 1 & 2 & \dots & k & k+1 & \cdots & n \\ k & k-1 & \dots & 1 & k+1 & \cdots & n \end{array}\right)

(i.e., {p_k} flips the top {k} pancakes). It is not hard to see that the diameter is between {n} and {2n}. Both lower and upper bound have been sharpened by a constant factor, and a natural conjecture is that {\mathrm{diam}~\mathrm{Cay}(S_n, P_n) / n} converges to some intermediate constant (to be determined, ideally).

Here is a natural model problem: estimate {|P^m|} for each fixed {m} as {n \to \infty}. Clearly {|P^m| \leq n^m}, and we should be able to detect some fall-off from this trivial upper bound.

Proposition 1 There is an integer sequence {(a_m)}, beginning

\displaystyle  1, 2, 6, 22, 90, 486, 3014, 22185, 182229, \dots

(not in the OEIS) such that {|P^m| \sim (a_m/m!) n^m} as {n \to \infty} with {m} fixed.

To define {(a_m)} it is useful to introduce the hyperoctahedral group {B_m = \{\pm1\} \wr S_m} (cf. burnt pancake sorting). You can think of {B_m} as the group of “signed” or “antipodal” permutations: permutations {\sigma} of {\{\pm1, \dots, \pm m\}} such that {\sigma(-x) = -\sigma(x)} for all {x}. There is a natural cover {B_m \to S_m} defined by forgetting signs. We lift {P_m} to {B_m} in a particular way: define {Q_m = \{q_1, \dots, q_m\}}, where

\displaystyle  q_k = \left(\begin{array}{ccccccc} 1 & 2 & \dots & k & k+1 & \cdots & n \\ -k & -(k-1) & \dots & -1 & k+1 & \cdots & n \end{array}\right).

Now define

\displaystyle  a_m = \# \{ q_{\sigma(1)} \cdots q_{\sigma(m)} : \sigma \in S_m \}.

I computed the initial terms in the sequence using this simple formula. The first nontrivial relations are

\displaystyle  q_1 q_2 q_4 q_3 = q_2 q_4 q_3 q_1

and its inverse (to verify I recommend using four asymmetric cards from a deck of cards), which together account for {a_4 = 22 < 24}.

To sketch the proof of Proposition 1, consider composing {m} random elements of {P_n}. You can think of this as a two-step process: randomly divide the stack {\{1, \dots, n\}} in {m} places, and then rearrange the resulting {m+1} pieces, possibly with some reversals. With high probability the points of division are all distinct, so they can be inferred from the result (this is true for {m} up to about {n^{1/2}}). The only question is whether the order can be inferred, and this is precisely what is defined by {a_m}.

It would be informative for the pancake problem to know more about the sequence {(a_m)}. I don’t know whether {a_m} can be computed more efficiently than the naive way above, e.g., whether there is a recurrence: if so that would really help a lot.

There is at least a submultiplicativity property that follows from the proposition:

Corollary 2 For {m, k\geq 1}, {a_{m+k} / (m+k)! \leq a_m / m! \cdot a_k / k!}. Hence there is a constant {c_1 > 0} such that

\displaystyle  a_m/m! < e^{-c_1m + o(m)}.

Proof: This follows from {|P^{m+k}| \leq |P^m| |P^k|}, dividing by {n^{m+k}}, and taking the limit. The second part follows from the subadditivity lemma and the data point {a_4 = 22 < 24}. \Box

I think a stronger bound should be true: there should be a constant {c_2 > 0} such that

\displaystyle  a_m < m!^{1-c_2 + o(1)}.

(One might then contemplate whether {n/(1 - c_2) + o(n)} is the diameter of the pancake graph, but that seems a little bold at this stage.)

Mixing time of a pseudorandom number generator

Today I was due to speak at the Isaac Newton Institute’s workshop on “Interactions between group theory, number theory, combinatorics and geometry”, which has obviously been canceled. I was going to speak about my work on the Hall–Paige conjecture: see this earlier post. Meanwhile Péter Varjú was going to speak about some other joint work, which I’d like to share with you now.

Consider (if you can consider anything other than coronaviral apocalapyse right now) the random process {(X_n)} in {\mathbf{Z}/p\mathbf{Z}} defined in the following way: {X_0 = 1} (or any initial value), and for {n\geq 1}

\displaystyle   X_n = aX_{n-1} + b_n. \ \ \ \ \ (1)

Here we take {a} to be a fixed integer and {b_1, b_2, \dots} a sequence of independent draws from some finitely supported measure {\mu} on {\mathbf{Z}}. As a representative case, take {a = 2} and {\mu = u_{\{-1, 0, 1\}}} (uniform on {\{-1, 0, 1\}}).

Question 1 What is the mixing time of {(X_n \bmod p)} in {\mathbf{Z}/p\mathbf{Z}}? That is, how large must we take {n} before {X_n} is approximately uniformly distributed in {\mathbf{Z}/p\mathbf{Z}}?

This question was asked by Chung, Diaconis, and Graham (1987), who were motivated by pseudorandom number generation. The best-studied pseudorandom number generators are the linear congruential generators, which repeatedly apply an affine map {x\mapsto ax+b \bmod p}. These go back to Lehmer (1949), whose parameters {a} and {b} were also subject to some randomness (from MathSciNet, “The author’s proposal for generating a sequence of `random’ 8 decimal numbers {u_n} is to start with some number {u_0\neq 0}, multiply it by 23, and subtract the two digit overflow on the left from the two right hand digits to obtain a new number {u_1}.”) The process {(X_n)} defined above is an idealized model: we assume the increment {b} develops under some external perfect randomness, and we are concerned with the resulting randomness of the iterates.

My own interest arises differently. The distribution of {X_n} can be thought of as the image of {a} under the random polynomial

\displaystyle  P_n(X) = X^n + b_1 X^{n-1} + \cdots + b_n,.

In particular, {X_n \bmod p = 0} if and only if {a} is a root of {P_n}. Thus the distribution of {X_n \bmod p} is closely related to the roots of a random polynomial (of high degree) mod {p}. There are many basic and difficult questions about random polynomials: are they irreducible, what are their Galois groups, etc. But these are questions for another day (I hope!).

Returning to Question 1, in the representative case {a = 2} and {\mu = u_{\{-1, 0, 1\}}}, CDG proved the following results (with a lot of clever Fourier analysis):

  1. The mixing time is at least

    \displaystyle (1+o(1)) \log_2 p.

    This is obvious: {X_n} is supported on a set of size {O(2^n)}, and if {n \leq (1-\varepsilon) \log_2 p} then {2^n \leq p^{1-\varepsilon}}, so at best {X_n} is spread out over a set of size {p^{1-\varepsilon}}.

  2. The mixing time is never worse than

    \displaystyle C \log p \log \log p.

  3. Occasionally, e.g., if {p} is a Mersenne prime, or more generally if {a} has small order mod {p} (or even roughly small order, suitably defined), the mixing time really is as bad as that.
  4. For almost all odd {p} the mixing time is less than

    \displaystyle 1.02 \log_2 p.

You would be forgiven for guessing that {1.02} can be replaced with {1+o(1)} with more careful analysis, but you would be wrong! In 2009, Hildebrand proved that for all {p} the mixing time of {X_n \bmod p} is at least

\displaystyle  1.004 \log_2 p.

Therefore the mixing time of {X_n \bmod p} is typically slightly more than {\log_2 p}. What is going on here? What is the answer exactly?

In a word, the answer is entropy. Recall that entropy of a discrete random variable {X} is defined by

\displaystyle  H(X) = \sum_{x : \mu(x) > 0} \mathbf{P}(X = x) \log \mathbf{P}(X = x)^{-1}.

Here are some basic facts about entropy:

  1. If {X} is uniform on a set of size {k} then {H(X) = \log k}.
  2. Entropy is subadditive: the entropy of the joint variable {(X, Y)} is at most {H(X) + H(Y)}.
  3. Entropy cannot be increased by a (deterministic) function: {H(f(X)) \leq H(X)}.

By combining the second and third facts, we have the additive version (in the sense of adding the variables) of subadditivity:

\displaystyle  H(X + Y) \leq H(X) + H(Y).

In particular, it follows from (1) that

\displaystyle  H(X_{m+n}) \leq H(X_m) + H(X_n),

and hence by the subadditive lemma {H(X_n)/n} converges. (We are thinking now of {X_n} as a random variable in {\mathbf{Z}}, not reduced modulo anything.) Call the limit {H = H(a, \mu)}.

It is easy to see in our model case {a=2, \mu = u_{\{-1, 0, 1\}}} that {H \leq \log 2} (because {X_n} has support size {O(2^n)}). If {H < \log 2}, then it follows that the mixing time of {X_n \bmod p} is strictly greater than {\log_2 p} (as {X_n} cannot approach equidistribution mod {p} before its entropy is at least {(1+o(1))\log p}).

Indeed it turns out that {H < \log 2}. This is "just a computation": since {H = \inf_{n\geq 1} H(X_n) / n}, we just need to find some {n} such that {H(X_n) / n < \log 2}. Unfortunately, the convergence of {H(X_n) / n} is rather slow, as shown in Figure 1, but we can take advantage of another property of entropy: entropy satisfies not just subadditivity but submodularity

\displaystyle  H(X+Y+Z) + H(X) \leq H(X+Y) + H(X+Z),

and it follows by a short argument that {H(X_n) - H(X_{n-1})} is monotonically decreasing and hence also convergent to {H}; moreover, unlike the quotient {H(X_n)/n}, the difference {H(X_n) - H(X_{n-1})} appears to converge exponentially fast. The result is that

\displaystyle  H / \log 2 = 0.98876\dots,

so the mixing time of {X_n \bmod p} is not less than

\displaystyle  H^{-1} \log p = (1.01136\dots ) \log_2 p.

(We can also deduce, a posteriori, that we will have {H(X_n) / n < \log 2} for {n \geq 72}, though it is out of the question to directly compute {H(X_n)} for such large {n}.)


Figure 1: The entropy difference rapidly converges. Predicted values are dashed.

This observation was the starting point of a paper that Péter and I have just finished writing. The preprint is available at arxiv:2003.08117. What we prove in general is that the mixing time of {X_n \bmod p} is indeed just {(H^{-1} + o(1)) \log p} for almost all {p} (either almost all composite {p} coprime with {a}, or alternatively almost all prime {p}). In other words, entropy really is the whole story: as soon as the entropy of {X_n} is large enough, {X_n \bmod p} should be close to equidistributed (with a caveat: see below). The lower bound is more or less clear, as above. Most of the work of the paper is involved with the upper bound, for which we needed several nontrivial tools from probability theory and number theory, as well as a few arguments recycled from the original CDG paper.

However, just as one mystery is solved, another arises. Our argument depends on the large sieve, and it therefore comes with a square-root barrier. The result is that we have to assume {H(a, \mu) > \frac12 \log a}. This is certainly satisfied in our representative case {a = 2, \mu = u_{\{-1, 0, 1\}}} (as the entropy is very close to {\log 2}), but in general it need not be, and in that case the problem remains open. The following problem is representative.

Problem 2 Let {a \geq 2} and let {\mu = u_{\{0, 1\}}}. Then {X_n} is uniform on a (Cantor) set of size {2^n}, so {H(a, \mu) = \log 2}. Show that the mixing time of {X_n \bmod p} is {(1+o(1))\log_2 p} for almost all primes {p}.

You might call this the “2–3–4–5 problem”. The case {a=2} is trivial, as {X_n} is uniform on {\{1, \dots, 2^n\}}. The case {a=3} is covered by our main theorem, since {\log 2 > \frac12 \log 3}. The case {a=4} is exactly borderline, as {\log 2 = \frac12 \log 4}, and this case is not covered by our main theorem, but we sketch how to stretch the proof to include this case anyway. For {a \geq 5} we need a new idea.