When you run this process at scale, the path-length histogram stops looking like a smooth fade-out and starts looking like a story with a loud, repeated ending. In our results, that ending clusters at 89 steps. This is not just a “common length.” It is a dominant one, a place where an unusually large share of paths terminate. A spike like that is what you get when randomness is being shaped by a constraint that tightens over time.

The rule that drives it is simple: the path is allowed to step to a neighboring point, but it is not allowed to step onto any point it has already visited. Early in a path’s life, almost every neighboring point is still unvisited, so the next move is usually safe. But every step leaves a permanent footprint, and those footprints accumulate into a kind of pressure field. As the path gets longer, it spends more time near its own history and more often finds itself surrounded by its own past. The number of safe neighboring choices shrinks, the number of dangerous choices grows, and the chance of death climbs.

What makes 89 stand out is that it sits right at the transition where “usually safe” becomes “usually crowded.” By the high double digits the path has drawn enough structure to create narrow corridors, partial cages, and tight turns that repeatedly bring it back near earlier segments. The path is not trying to go anywhere, but its own history has started to shape the space around it. At that point, a single unlucky choice is enough to end everything, and the statistics show that this happens most often right around 89.

This chart treats 89 steps as a single unit of time and asks a simple question: once the walk reaches the peak-length scale, how quickly does the population thin out as we move to 2 times 89, 3 times 89, 4 times 89, and so on. Each point on the x-axis is a path-length bucket that is exactly one more 89-step interval than the last, and the curve shows how the counts change across those repeated intervals.

The first feature is the obvious one: the 89 bucket is not just large, it is oversized compared to its neighbors. Normalized to the zero bucket, the 89 bucket rises to about 140 percent, which is why it reads as a true peak rather than a gentle hump. But the more important story begins immediately after that peak. From 89 onward, the ratios between consecutive buckets settle into a remarkably consistent decay. The percent column in the raw table is the key: it is the survival ratio from one 89-step bucket to the next. After the early transition, that ratio hovers around 51 to 53 percent for a long stretch. In plain terms, once you are past the peak, each additional 89 steps cuts the remaining population roughly in half.

That is why this chart behaves like a half-life curve. It is not describing a single dramatic event; it is describing a repeating rule. The walk accumulates history, and history creates constraints, and the cost of extending life by one more 89-step interval is that about half of the candidates will fail before reaching the next interval. What you are seeing in the long middle section is the system falling into an almost geometric decay pattern, where the tail of the distribution is governed by a near-constant hazard per 89-step block.

This chart shows the same distribution two different ways, and the contrast is the point. The blue curve, percent of max, answers “how common is each bucket compared to the single most common bucket.” It surges up to a sharp early peak, then falls fast and keeps falling. That is the shape of a system where most outcomes concentrate near the front. The walk quickly enters a regime where surviving to the next bucket becomes harder, so the frequency collapses and never really recovers.

The orange curve, percent of sum, tells the same story from the opposite angle. It is the running total, how much of the entire population you have captured as you move from left to right. It climbs steeply at the start, then flattens out and creeps toward 100 percent. That steep rise means the early buckets contain the bulk of all paths. The slow final crawl means the rest of the x-axis is a very long right tail, not because it holds a lot of probability, but because it spans a lot of space.