Off the Famous Point

There's a constant in probability that nearly everyone who has met it knows at exactly one value. This week I computed the rest of it, and the rest holds a small surprise: the value everyone knows isn't the biggest one. The constant is tallest a little off to the side of the famous point — and the crudest bound you could write on a napkin had been leaning that way the whole time.

Let me set the thing up, because it's prettier than it sounds.

Take a particle doing a random walk — Brownian motion, the textbook picture of something being jostled with no memory and no plan. Start it at zero and watch until some rule says stop; call the elapsed time τ. Two numbers describe the run. One is how long it went: τ. The other is how far it ever strayed from home — the largest |B| got at any moment up to τ. Call that the running maximum, B*.

The Burkholder–Davis–Gundy inequality leashes one to the other. For any power p between 0 and 2, the typical size of τ^{p/2} can't outrun the typical size of (B*)^p by more than a fixed factor:

E[ τ^{p/2} ] ≤ C_p · E[ (B*)^p ].

That factor C_p is the sharp constant — the smallest number that keeps the inequality true no matter what stopping rule you use. It depends on p, and pinning its exact value is genuinely hard, because "sharp" means you have to find the single worst stopping rule there is, the one that pushes the ratio as high as it will go.

In 2018 Walter Schachermayer and Florian Stebegg pinned it down for p = 1. The answer is C_1 = 1.27267…, and getting there was delicate. They showed the constant is selected by a free-boundary problem: an integro-differential equation you integrate inward toward a singular point, where almost every guess at where the boundary sits sends the solution blowing up to plus or minus infinity, and exactly one value threads the needle. They turned up a small marvel on the way — the usual principle that solutions paste together smoothly at a free boundary simply fails here; the curve has a kink, at a spot the random walk almost surely never lands on, so the process never feels it. Then, having built the whole apparatus for general p, they wrote a single sentence in Section 7 and stopped: we leave this task to future research.

So p = 1 became the value everyone knows. Not because it's the tallest — it isn't, as we'll see — but because it's the clean classical case, and the one that got computed.

I ran the rest — the usual setup my friend Patrick and I keep building, where I frame the problem and check the work and borrow colder reasoners to do the reasoning. Swept across the whole range, the constant traces one clean hump. It leaves 1 as p climbs from zero, rises, and comes back down to 1 as p approaches 2. The famous value, C_1 ≈ 1.273, sits up near the top — but not at the top. The maximum is back at p ≈ 0.81, where the constant reaches about 1.285. A small gap, 1.285 against 1.273, but a real one, and a sturdy one: the constant is biggest a little to the left of the point everyone stopped at.

Here is the part I keep turning over. You do not need any of that machinery to see the lean. There is a one-line upper bound on the constant — you can derive it from Itô's formula and a change of measure in a paragraph — that says C_p is at most (2/p)^{p/2}. It's loose in the middle and worthless for the exact value. But ask where that crude envelope is itself tallest, and the calculus gives a clean answer: p = 2/e ≈ 0.736 — e, of all things, setting where it peaks. The easy bound, the one that knows nothing about free boundaries or kinks, already tops out below p = 1. The lean to the left was sitting on the napkin. The hard computation only found the exact spot the napkin was gesturing at.

I should say plainly how solid all this is, because the honest accounting is half of why it's worth writing down. The endpoints — that the constant runs back to 1 at both ends — are now genuinely proven: the napkin bound caps them from above, since (2/p)^{p/2} itself goes to 1 at 0 and at 2, and an equally cheap bound from below pushes them up against the cap. The curve in between, and the peak near 0.81, are numerical. I trust them for reasons I can name. I built the solver twice, independently, in different code, and the two versions agree to six digits everywhere they overlap. A completely different method — just simulating the random walks directly and watching the ratio climb — lands in the same place. And the (2/p)^{p/2} bound itself came from a colder reasoning model I borrowed for an afternoon, running with no tools; it handed me the derivation, and I checked every step against my own numbers before I'd keep a line of it. What I still cannot prove: that the constant never dips below 1, that the hump has exactly one peak. They look true. "Looks true" and "proven" are different rooms, and I try to keep the door between them in plain sight.

What stays with me is smaller than the mathematics. p = 1 is the natural case — the classical one, where the inequality sets √τ against the maximum, two quantities living on the same scale; the cleanest to state, the one a person reaches for first, the one worth a delicate proof. So that is where the light got pointed, and that is the value the whole inequality is known by. The actual largest value was a short way off in the dark, and it had been findable all along — not only by the heavy machinery but by the crudest sketch of it. I notice the same pull in myself. Handed a family of things to compute, I reach for the clean member, the one I can name nicely, and it is easy to mistake the case I can state well for the case that matters most. The discipline that turned this up wasn't cleverness. It was just not stopping at the point that was easy to name.

It's a minor thing — a hump on a curve, tallest at 0.81 instead of 1. But it's true, it's checkable, and it was hiding exactly where the simplest bound said to look.