Each “like” turns a spigot off, stopping water from flowing through one pipe in a grid overhead. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. [Emphasis added] The better your “likeability,” the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).

This system models a type of phase transition known as percolation.

The full post is here:

I highlighted above a key phrase “If it can find a path to the bottom, you get soaked.” What I didn’t say, but should have is that the water was being forced through the pipes, not just dropping down due to gravity. This is a very important point since our phases and phase transition changes dramatically if we just let gravity do the work. In the case of the water being “forced,” it can travel back up pipes if it helps it find its way out and onto your head, but in the case when only gravity is present, it falls down the pipes. To facilitate gravity, we’ll turn the pipes 45 degrees, and if we insert water at a single point on top, it could look like this:

This setup is a different problem called directed percolation. It also has a phase transition, but one that is different in some fundamental ways from regular percolation.

Before we explore its stranger properties, we can ask, “At what likability threshold do you remain dry?” Well, this happens to have a transition chance of 35.53%!2 This system is a lot more generous, keeping you dry even when a majority of people dislike you. This number comes from numerical computations which have been done rather precisely, and we can even compute it ourselves. In fact, you can see this clearly with this plot

Notice that as we make the system bigger and bigger, the chance of getting soaked less than 35.53% increases and above it, it decreases. This is the same kind of hallmark of a phase transition as we saw in our previous case.

We can also look at the water as it flows down the system to see the clusters that make it from top to bottom

There is still a fractal-looking pattern at the transition point. With all of these similarities with the regular percolation problem from the last post, what is different? And why is that plot so long and skinny? If gravity wants to pull you down, is that somehow altering the motion down, making it distinct from the motion left or right?

Well, if you go back to the two plots above, you’ll notice a few things that really make them differ from the percolation plots. In the fine print of the first, I’ve noted that the vertical distance is L^1.58, so for a horizontal size of 40, the vertical size is roughly 340! That is definitely not a square. And in the second plot, there appears to be more vertical distance than horizontal distance. What is special about this 1.58 number3? It turns out, it’s a critical exponent in this problem, a universal aspect of directed percolation, that distinguishes it from regular percolation. We will call it z = 1.58 the dynamical critical exponent since it is revealed as water flows down in time (dynamically). This dynamical exponent z can reveal itself by looking at these “long and skinny” setups, but be masked by the square setup.

Universality and the finite size of our system

One thing we took away in the previous post was that we lose any sense of scale at this type of phase transition4. But whenever we have “only” thousands of pipes, the size of the system provides a scale! This is the main reason why we begin to see smooth curves and not sharp jumps in quantities. If the system of pipes were infinite (and we had infinite time for the water to go down the pipes), the probability you get soaked would be 100% less than the 35.53% likeability and 0% more than 35.53% likeability. For physical systems, the finite size is often not a huge issue since the scale is closer to the 10²³ atoms present in macroscopic systems, and so even things that are technically smooth curves look very sharp.

The problem of size becomes more severe with directed percolation because horizontal and vertical distances start behaving differently thanks to gravity. In this case, if we lay out our nice grid of 10 × 10, 20 × 20, or 30 × 30, we start to notice that the likeability threshold where you stop getting soaked, seems to depend on the size of the system more than before. In actuality it doesn’t, but for these small sizes, you are definitely getting soaked well into the so-called “Dry Phase” we previously labeled. This is seen in the red curves here where each bigger square has a curve underneath the last:

Gravity has done something to the system. Flowing down is different from flowing left or right. In fact, if we flow down by some amount h and over to the right by some distance w, then at the directed percolation transition point

The amount water flows down is related to how far it flows to the right or left by this weird, fractional power of w. This 1.58 is z, our new dynamical critical exponent, which is a universal feature of directed percolation5. It tells us that if we make a system 30 pipes wide, it should extend roughly 30^1.58 ≈ 216 pipes in height to begin picking out the phase transition effectively. The blue curves in the above plot show this and notice how they all converge on one point; that point is the phase transition. It is revealed by small sizes! To understand why, just think about how the curves are changing as we make the system bigger and bigger.

The red curves will still converge to the phase transition, but it takes larger system sizes for it to reveal itself. This is related to the property that at the phase transition there is no longer a sense of scale, but away from the transition, the vertical scale of clusters could be so large that our puny 60-by-60 grid cannot even begin to reveal it. So if we sit at say a likeability of 0.4 in the 60-by-60 grid, we can say that the vertical size of a typical cluster is most likely more than 60.

A different phase transition but connections to new types of physics

This “gravity mode” for our game show we may call “easy mode” since it requires less of the audience to like you, but the implications here are wide. This type of phase transition has been seen in many kinds of local dynamics where there is a preferred configuration or state. These called an absorbing state phase transitions, and they are a property of certain random dynamical systems. Gravity has provided the distinction here, but more generically, causality and time itself provide that direction, leading to dynamics that obey the same universality as directed percolation.

What is the Percolating Phase?

Imagine you’re on a game show called Are You Likeable? (the least-likeable game show). The rules of the game are simple:

1. You stand on the stage and try to win over the audience.

2. Each audience member votes whether they like you or not.

3. But the twist: votes aren’t tallied—they control a system of pipes above your head.

That system of pipes looks something like this2

Each “like” turns a spigot off, stopping water from flowing through one pipe in a grid overhead3. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. The better your “likeability,” the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).

This system models a type of phase transition known as percolation4.

But where is the phase transition?

Aside from asking when my game show will be green-lit5, we can ask: When are you most likely to get wet? If the audience is huge, and fewer than 50% of them like you, it’s nearly guaranteed that the water will find a path—you’ll be soaked. But if more than 50% like you, chances are good that you’ll stay dry.

This is a phase transition6: a sudden change in the system’s behavior. For a moderately sized audience, it looks something like this:

Your probability of getting soaked forms a curve that sharpens up with increasing audience size—becoming a near step at 50%. That is known as the percolation threshold.

This is hard to visualize, though; luckily this problem admits some very nice pictures. For instance, here is the problem with a large number of pipes:

If a pipe is blue, water is in it, and all the blue clusters flow down from the top. Notice what happens around 50%: even though spigots are randomly being turned off, the flow from top to bottom is entirely stopped around this value. Something is happening within the system that is allowing water to pass through on one side of the transition and not the other.

A closer look at the transition

To dig deeper, we simulate what happens at this threshold. Each spigot is either open or closed (randomly determined). If we visualize the grid (say 1024×1024 spigots), it looks like visual static: black and white dots with no obvious pattern7:

But now let’s color each connected cluster of open spigots—where water could flow and fill up this section of pipes. Suddenly, structure emerges. Some clusters are small, others large. If one spans from top to bottom, water flows and we’re in the percolating phase. If not, we’re in the non-percolating phase. At the transition (within the static above, we get this for the twelve largest clusters:

At exactly the percolation threshold (50% for the pipes above), there’s no single dominant cluster, but also no clear typical size. Instead, there’s a wide distribution of cluster sizes. The critical state behaves differently than either phase.

Scale Invariance: A Hallmark of Criticality

Let’s zoom out. Suppose we double the grid to 2048×2048.

The largest clusters are definitely larger—the largest here is 400,000 pipes/spigots large, while for the previous 1024×1024 case the largest was 180,000 large—but the pattern still looks… the same. We doubled the size, but if we slightly blur our vision, we cannot distinguish these two plots (even though one is quadruple the area of the other). Look at 512×512—even that looks similar:

You would be hard-pressed to say which one is larger if you blurred your eyes. This problem is apparent even down to 256×256 or 128×128.

This is called scale invariance—there is no characteristic length scale at the phase transition. It’s one of the defining features of what are known as second-order phase transitions.

It also explains why, at the threshold, you have a 50% chance of getting soaked. The largest cluster might span the system, but it might just as well fall short. There’s no guarantee either way.

Fractals in the Flow

These clusters within the above pictures don’t look like regular 2D structures or even 1D lines; they are, in fact, fractals. They aren’t exactly self-similar but they do behave the same at different scales. They fill space with a fractal dimension: not quite 1D, not quite 2D. In two-dimensional percolation, the clusters have a dimension of 91/48 ≈ 1.896—a universal number shared by all systems in this class, regardless of lattice type or other microscopic details.

This is part of the beauty of percolation: It shows us visually the underlying mathematical structure and even reveals some universality of phase transitions.

Why This Matters

Percolation is just one example, but it captures the essence of what physicists mean when they talk about “phases of matter.” It isn’t always about exotic particles or extreme temperatures turning gas into plasma. Sometimes, it’s about whether a liquid can find its way through a series of pipes. It’s about symmetry, structure, and emergence.

You’ve experienced water’s phases: ice, liquid, steam. But nature offers many more—some with no neat label like “solid” or “gas.” The theory of phase transitions explains them. Percolation is a window into that wider world.