Faster Is Not Nearer

Imagine a machine that randomly rotates and reflects a labeled regular $n$-sided polygon. At each point in time, there are $2n$ possible states to which the polygon can belong. Each possible move has its own clock that rings at a certain random rate. When a clock rings, the machine performs that move. Assume that each move has the same rate as its inverse. Starting from a fixed state, the distribution of the polygon's state approaches an equilibrium where all states are equally likely. 

Now speed up some of the clocks while making sure that the rates for moves that are inverses of each other remain equal. This causes the machine to make moves more frequently, so surely it should approach equilibrium at least as quickly. Well, not so fast! It turns out that, in a certain sense, increasing the rate can actually cause the distribution at a given time to move further from equilibrium. This is analogous to the familiar image of a person caught in quicksand. Moving more vigorously trying to escape can cause the person to sink more, getting further from freedom. 

In the polygon story, whether the extra motion helps or hurts depends on how we measure the distance from equilibrium. A measurement based on powers of 2 or 4 never gets worse under these changes. However, a measurement based on a power of 4.001 can. In Proof of the Lyons–White Conjecture, mathematicians at Axiom Math explain this distinction precisely.

The rotations and reflections of a regular $n$-gon form the dihedral group $D_n$, which has $2n$ elements. These are the states of our random walk. Let $P_t(g)$ be the probability of being in state $g$ at time $t$. For each fixed $g$, the limit of $P_t(g)$ as $t$ tends to $\infty$ is $1/(2n)$. Note that at time $0$, the walk is at the initial state, which is the identity element of the group, with probability $1$. This initial distribution is very far from equilibrium because all of the probability mass is concentrated on a single state. For a fixed time $t$, we can measure the distance from equilibrium via the quantity

$$d_p(t)=\left(\sum_{g\in D_n}\left|P_t(g)-\frac{1}{2n}\right|^p\right)^{1/p}.$$

This measurement depends on the parameter $p \geq 1$. If $p=1$, we add the absolute discrepancies. At $p=2$, we square them, add, and take a square root. Larger exponents place relatively more emphasis on the larger discrepancies. The limiting case $p=\infty$ records only the largest one. 

The main question concerns whether increasing the clock rates can increase $d_p(t)$ for some fixed $t$. If this can never happen for any fixed $t$, then we say the pair $(D_n,p)$ is rate-monotonic. The first surprising fact is that some pairs are not rate-monotonic. Fundamentally, this is because increasing the rate of a single move will increase the chance of applying that move, but it will also increase the chance of subsequently undoing the same move. The second surprising fact is that rate-monotonicity can have an extremely sensitive dependence on $p$.   

Russell Lyons and Graham White had already established that $(D_n, p)$ is rate-monotonic when $p=2$ and when $p=\infty$ (in fact, they proved a much more general result for arbitrary finite groups). Other exponents behaved differently. They found dihedral examples where increasing a rate made the distribution farther from uniform, including examples at exponents very close to 4 or 6. Yet they found no such examples for $p=4$ or $p=6$. 

The first main result in Axiom’s paper says that $(D_n,p)$ is rate-monotonic whenever $p$ is an even positive integer. They also prove a converse: for every finite $p \geq 1$ that is not an even integer, there is some $n$ for which $(D_n, p)$ is not rate-monotonic. Thus, the even integers $2,4,6,8,\ldots$ are quite special. 

Note that the theorem does not say that every walk exhibits the counter-intuitive lack of monotonicity when $p$ is, say, $4.001$. It says that if we choose $p$ to be $4.001$, then there is some dihedral group, some choice of rates, and some observation time for which increasing some of the rates causes the distance to equilibrium to increase.

The proof that $(D_n, p)$ is rate-monotonic when $p$ is an even positive integer applies tools from Fourier analysis. The cyclic group of rotations supplies a natural collection of frequencies. By working with these frequencies, the authors reduce the essential noncommutative calculations to $2 \times 2$ matrices. Positivity of these matrices gives bounds between the Fourier coefficients of the quantities appearing in the proof. The next task is to turn those bounds into a comparison of distances. 

For a positive integer $m$, an even power can be rewritten as a square:

$$\sum_j |f_j|^{2m} = \sum_j |f_j^m|^2. $$

Now Parseval’s identity becomes available. A sum of squared magnitudes can be computed on the Fourier side. Meanwhile, the Fourier coefficients of the integer power $f^m$ expand into finite sums of products. When the original coefficients are bounded in magnitude by nonnegative Fourier coefficients, these products can be bounded term by term. The result is a comparison of the corresponding norms. 

The remaining argument connects this comparison to the clocks. The authors differentiate the distance, raised to the $2m$-th power, with respect to a jump rate. Their matrix inequality and Fourier comparison control the crucial terms, showing that increasing a symmetric rate cannot make the distance larger. Increasing the rates one at a time then gives the theorem.

For each $p \geq 1$ that is not an even integer, the authors identify a negative Fourier coefficient in the nonlinear expression governing how the $\ell^p$-distance changes. They utilize this to construct a small perturbation involving two frequencies. The resulting expression responds to an increased reflection rate in the wrong direction. 

There is still a substantial obstacle. A construction of a convenient Fourier coefficient need not describe a genuine random walk. Indeed, it coefficients must ultimately come from nonnegative jump rates with enough allowed moves to generate the whole dihedral group. The authors show how to realize their construction, after suitable adjustments and scaling, as the centered distribution of such a walk. 

The mathematical quicksand has now been constructed, and one obtains a system in which speeding up one particular motion pulls the distribution farther from its goal.

This division between even and non-even exponents also appears in the classical Hardy–Littlewood majorant problem, which asks how bounds on Fourier coefficients control norms. The paper connects that analytic distinction to a probabilistic one.

There is a completely analogous way of defining a random walk on any finite group $G$, which depends on a choice of rate for each element of the group. We assume once again that each element and its converse are assigned the same rate. Thus, for any $p \geq 1$, one can again ask whether the pair $(G, p)$ is rate-monotonic. The first theorem of the article generalizes to a broader family of groups called inversion extensions of finite abelian groups; these generalize the dihedral groups. The paper shows that if $G$ is any inversion extension of a finite abelian group, then $(G, p)$ is rate-monotonic for every even positive integer $p$. The paper also proposes an even broader conjecture that if $G$ is a finite group with an abelian subgroup of size $|G|/2$, then $(G, p)$ is rate-monotonic for every even positive integer $p$.

The paper has a second layer beyond the mathematics itself. AxiomProver generated Lean/Mathlib formalizations of the central results. These formal files are not meant to replace the paper. They serve as a machine-checkable record that the main deductions can be carried out with the hypotheses made precise.

The mathematical content still needs a human narrative. A formal proof can certify that a chain of implications is valid, but it does not by itself explain why the problem splits so sharply between even and non-even exponents. That explanation comes from the surrounding theory, including Fourier analysis on dihedral groups and positivity of certain $2 \times 2$ matrices. These ideas reveal why the positive even integers provide firm ground, while nearby exponents can behave like quicksand.

Paper: Colin Defant and Ken Ono. Proof of the Lyons–White conjecture. 

arXiv: https://arxiv.org/abs/2608.27708v1

Code repository: https://github.com/AxiomMath/LyonsWhite