‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions

The week before Christmas 2025, five mathematicians were holed up in a classroom at ETH Zurich.

2025 年圣诞节前一周，五名数学家躲在苏黎世联邦理工学院的一间教室里。

The mood was electric: They were this close to a career-defining breakthrough.

大家的情绪激动不已：他们距离职业生涯的突破如此之近。

The group — consisting of then-postdocs Sahar Diskin and Philip Easo , graduate student Ritvik Ramanan Radhakrishnan , Benny Sudakov , and Vincent Tassion — was perfecting a solution to one of the biggest open problems in percolation theory, the study of flow in a network.

该小组由当时的博士后 Sahar Diskin 和 Philip Easo、研究生 Ritvik Ramanan Radhakrishnan、Benny Sudakov 和 Vincent Tassion 组成，正在完善渗透理论中最大的开放问题之一（网络流研究）的解决方案。

Percolation captures a vast array of phenomena, but the prototypical examples involve fluids, like hot water seeping through a bed of coffee grounds.

渗滤捕捉到了大量的现象，但典型的例子涉及流体，比如热水渗入咖啡渣床。

Diskin, Easo, Radhakrishnan, Sudakov, and Tassion were trying to work out something fundamental about how graphs — networks of points connected by lines, or edges — can be taken over by large connected areas, the equivalent of pools of fluid.

Diskin、Easo、Radhakrishnan、Sudakov 和 Tassion 试图解决一些基本问题，即图形（由线或边连接的点组成的网络）如何被大型连通区域（相当于流体池）接管。

The group had glimpsed a simple argument that could deal with a huge variety of graphs at once.

该小组瞥见了一个可以同时处理大量图表的简单论证。

“We almost didn’t believe it at first,” Radhakrishnan said.

“一开始我们几乎不相信，”拉达克里希南说。

They raced to confirm each detail, eager to get their idea down before it shimmered away — and heedless of the holiday.

他们争先恐后地确认每一个细节，渴望在想法付诸实践之前把它写下来——而且不顾假期。

“I’m not sure the girlfriends and the families were as happy as we were.

“我不确定女朋友和家人是否和我们一样快乐。

But we were all very happy at that moment,” Diskin said.

但那一刻我们都很高兴，”迪斯金说。

“It’s really rare that you’re able to hit something that feels so big and so meaningful.”

“你能够击中感觉如此重大且如此有意义的东西，这真的很罕见。”

They worked through the night.

他们通宵工作。

By the morning of December 17, exhilarated from the effort, they were convinced their idea was correct.

到了 12 月 17 日早上，他们对这一努力感到兴奋不已，并确信自己的想法是正确的。

By Christmas, they’d nailed down a proof.

到了圣诞节，他们已经确定了证据。

They had answered a decades-old question about how fast a percolation network floods as you open it up to fluid flow.

他们回答了一个几十年前的问题：当渗透网络向流体流动时，其洪水泛滥的速度有多快。

“I find great joy in this proof,” said Asaf Nachmias of Tel Aviv University, who studies percolation theory and probability.

“我对这个证明感到非常高兴，”特拉维夫大学研究渗透理论和概率的阿萨夫·纳赫米亚斯说。

“It’s stunning.”

“太棒了。”

From left: Sahar Diskin, Ritvik Ramanan Radhakrishnan, Philip Easo, Vincent Tassion, and Benny Sudakov take a group selfie after completing their paper on supercritical sharpness.

左起：Sahar Diskin、Ritvik Ramanan Radhakrishnan、Philip Easo、Vincent Tassion 和 Benny Sudakov 在完成超临界清晰度论文后合影。

Percolation can describe many kinds of flow: the spread of a virus through a city, gas passing through a filter, or the propagation of a wildfire.

渗透可以描述多种流动：病毒在城市中的传播、气体通过过滤器的传播或野火的传播。

But its original inspiration was coal.

但它最初的灵感是煤炭。

In the 1940s, the scientist Rosalind Franklin — now famous for her work on the structure of DNA — was employed at the British Coal Utilization Research Association (BCURA), trying to understand the intricate properties of coal, charcoals, and graphite.

20 世纪 40 年代，科学家罗莎琳德·富兰克林（Rosalind Franklin）——如今因其在 DNA 结构方面的研究而闻名——受聘于英国煤炭利用研究协会 (BCURA)，试图了解煤炭、木炭和石墨的复杂特性。

Scientists knew that coal was studded with tiny holes, but they didn’t know why some types of coal allowed fluids to pass through them, while others were impermeable.

科学家们知道煤炭上布满了小孔，但他们不知道为什么某些类型的煤炭允许液体通过，而另一些则不渗透。

By submerging coal in a variety of fluids, Franklin was able to measure the typical size of its holes, as well as the amount of variation.

通过将煤浸入各种液体中，富兰克林能够测量其孔的典型尺寸以及变化量。

About a decade later, the researchers Simon Broadbent and John Hammersley — wanting to understand the carbon filters in gas masks — developed a mathematical model .

大约十年后，研究人员西蒙·布罗德本特和约翰·哈默斯利想要了解防毒面具中的碳过滤器，他们开发了一个数学模型。

Their idea was simple.

他们的想法很简单。

Take a grid of evenly spaced points (also called a lattice) and a coin.

取一个由均匀分布的点组成的网格（也称为格子）和一枚硬币。

For each pair of neighboring points, flip the coin.

对于每对相邻点，抛硬币。

If it lands on heads, connect the points with an edge.

如果它落在正面，则用边缘连接这些点。

Fluid can flow between these points.

流体可以在这些点之间流动。

If the coin lands on tails, the flow is blocked.

如果硬币反面落地，则流动被阻塞。

Repeat this procedure for every pair of points.

对每对点重复此过程。

How far does the fluid go?

液体能流多远？

The answer depends on the probability that your coin lands on heads, which can range from 0% to 100%.

答案取决于硬币正面朝上的概率，范围从 0% 到 100%。

When the probability is low, fluid can flow through only a few channels, and so it collects in small, isolated puddles.

当概率较低时，流体只能流经几个通道，因此它会聚集在小的、孤立的水坑中。

But once the probability passes a threshold called the critical probability, the lattice suddenly opens up.

但一旦概率超过称为临界概率的阈值，晶格就会突然打开。

Fluid can travel extensively through the system.

流体可以在系统中广泛流动。

The exact value of the critical probability changes depending on the shape of the lattice — a square lattice has a different critical probability than a triangular one, and a 3D lattice has a different critical probability than a 2D one.

临界概率的确切值根据晶格的形状而变化——正方形晶格与三角形晶格具有不同的临界概率，3D晶格与2D晶格具有不同的临界概率。

But as you move above that critical value, you’ll see a phase transition, like liquid water turning to ice.

但当你超过这个临界值时，你会看到相变，就像液态水变成冰一样。

On a finite graph, crossing the critical probability means the network will become dominated by one large sea of fluid.

在有限图上，跨越临界概率意味着网络将被一大片流体所主导。

On an infinite graph — an abstraction where the graph extends forever in all directions — one or several infinite seas will dominate.

在无限图上——图形向各个方向永远延伸的抽象——一个或几个无限的海洋将占主导地位。

Physicists quickly realized that through percolation, they could learn about melting and freezing, as well as other phase transitions like magnetization.

物理学家很快意识到，通过渗透，他们可以了解熔化和冻结，以及磁化等其他相变。

“Phase transitions in physics are very hard to study rigorously,” Easo said.

“物理学中的相变很难严格研究，”伊索说。

“Percolation is like the caricature.

“渗透就像漫画一样。

So people often try to study that first, and then tools trickle down.”

因此，人们通常会首先尝试研究这一点，然后工具就会慢慢涌现。”

For scientists who had long been stymied by complicated real-world phase transitions, “it was catching the essence in a very simple setup,” said Itai Benjamini of the Weizmann Institute for Science.

魏茨曼科学研究所的伊泰·本杰明 (Itai Benjamini) 表示，对于长期被复杂的现实世界相变困扰的科学家来说，“它通过一个非常简单的设置抓住了本质”。

“A lot of things that could cloud the issue were removed.”

“很多可能使问题变得模糊的东西都被删除了。”

Itai Benjamini, along with his collaborator Oded Schramm, made early progress studying the percolation of transitive graphs.

Itai Benjamini 和他的合作者 Oded Schramm 在传递图渗透的研究中取得了早期进展。

For decades, researchers worked to quantify the percolation phase transition.

几十年来，研究人员致力于量化渗滤相变。

They wanted to know exactly how quickly pools of fluid can grow as you increase the probability that your coin lands on heads.

他们想确切地知道，当硬币正面朝上的概率增加时，液体池会以多快的速度增长。

Many predicted that the pools grow very fast — that below the critical probability, puddles are tiny, and above it, a single ocean covers almost everything.

许多人预测水池增长得非常快——低于临界概率，水坑很小，而高于临界概率，一片海洋几乎覆盖了一切。

This prediction is called the sharpness conjecture.

这种预测称为锐度猜想。

When sharpness was proved on lattices in the 1980s — by two independent groups, one in New Jersey and one in Moscow — it was “foundational,” said Tom Hutchcroft of Princeton University and the California Institute of Technology, who was Easo’s doctoral adviser.

埃索的博士生导师、普林斯顿大学和加州理工学院的汤姆·哈奇克罗夫特 (Tom Hutchcroft) 表示，20 世纪 80 年代，两个独立小组（一个在新泽西，一个在莫斯科）在晶格上证明了清晰度，这是“基础性的”。

Knowing sharpness, mathematicians can deduce a lot about the structure of the flooded portion of the network — in particular, that it looks very similar to the underlying lattice.

了解了清晰度，数学家可以推断出很多有关网络被淹没部分的结构的信息，特别是它看起来与底层晶格非常相似。

So when Benjamini and his colleague Oded Schramm plotted an expedition to bring percolation to new types of networks, it seemed natural to wonder if sharpness would go with them.

因此，当本杰明尼和他的同事奥德·施拉姆 (Oded Schramm) 策划一次探险活动，将渗透引入新型网络时，人们自然会想知道清晰度是否会随之而来。

In 1996, Benjamini and Schramm wanted to study percolation in a much larger class of graphs, called transitive graphs.

1996 年，Benjamini 和 Schramm 想在更大的一类图（称为传递图）中研究渗透。

To understand what a transitive graph is, imagine the graph as a network of roads on a flat, desolate landscape.

要理解什么是传递图，请将图想象为平坦、荒凉的风景上的道路网络。

If you want to know where you are on these roads, the only landmarks are the intersections.

如果您想知道自己在这些道路上的位置，唯一的地标就是十字路口。

But if the graph is transitive, all the intersections look similar — to figure out where you are, you’ll need GPS or a compass.

但如果图是传递性的，所有的交叉点看起来都很相似——为了弄清楚你在哪里，你需要 GPS 或指南针。

A square lattice is one example of a transitive graph: Every intersection consists of four edges meeting at right angles.

方格是传递图的一个示例：每个交点都由四个以直角相交的边组成。

But there are many kinds of transitive graphs — simple loops (below left) and infinitely expanding “trees” (below right), as well as ones that are almost impossible to visualize.

但传递图有很多种——简单的循环（左下图）和无限扩展的“树”（右下图），以及几乎不可能可视化的图。

Many transitive graphs represent objects from other mathematical subfields, like algebra or geometry.

许多传递图表示来自其他数学子领域的对象，例如代数或几何。

Benjamini was intrigued by these interdisciplinary possibilities — he hoped the percolation process would reveal insights into the graph itself.

本杰明对这些跨学科的可能性很感兴趣——他希望渗透过程能够揭示对图表本身的见解。

“You have a stage, which is geometry, and a dancer, which is the random process,” he said.

“你有一个舞台，这是几何学，还有一个舞者，这是随机过程，”他说。

By watching the dancer, he hoped to learn more about the stage.

通过观看舞者，他希望更多地了解舞台。

Over the next decade, Benjamini, Schramm, and their colleagues published a flurry of results on the percolation of transitive graphs.

在接下来的十年里，本杰明、施拉姆和他们的同事发表了一系列关于传递图渗透的结果。

They proved that, for a class of infinite transitive graphs, percolation exhibits a phase transition as you open up edges to flow: Small, isolated pools of fluid suddenly coalesce into an infinite web of connected rivers.

他们证明，对于一类无限传递图，当你打开流动的边缘时，渗透会表现出相变：小的、孤立的流体池突然合并成一个由相互连接的河流组成的无限网。

But they still didn’t know how fast that transition happened.

但他们仍然不知道这种转变发生的速度有多快。

Below the critical point, how big and how numerous were the pools?

在临界点以下，水池有多大、数量有多少？

Above it, was the infinite web a meadow crisscrossed with streams — or was it more like an ocean, swamping the entire graph?

在它的上方，无限的网络是一片溪流纵横交错的草地——还是更像是一片海洋，淹没了整个图表？

Benjamini and Schramm suspected that a version of the sharpness conjecture was true on all infinite transitive graphs.

本杰明尼和施拉姆怀疑锐度猜想的一个版本在所有无限传递图上都是正确的。

That conjecture could be broken down into two separate problems.

这个猜想可以分解为两个独立的问题。

The “subcritical” half — addressing what happens below the critical point — was completed in 2007, by Tonći Antunović and Ivan Veselić .

“亚临界”部分——解决临界点以下发生的情况——由 Tonći Antunovic 和 Ivan Veselić 于 2007 年完成。

Their work showed that here, pools of fluid are tiny and far apart.

他们的研究表明，这里的液体池很小而且间隔很远。

Even a hair below the critical point, the system looks more like Arizona than Minnesota.

即使比临界点低一点点，该系统看起来也更像亚利桑那州而不是明尼苏达州。

You have a stage, which is geometry, and a dancer, which is the random process.

你有一个舞台，这是几何学，还有一个舞者，这是随机过程。

Itai Benjamini, Weizmann Institute for Science

伊泰·本杰明，魏茨曼科学研究所

The “supercritical” half of the conjecture — which deals with probabilities above the critical threshold — seemed harder.

该猜想的“超临界”部分——处理高于临界阈值的概率——似乎更难。

Here, the landscape should be made up of possibly many seas, each infinitely large.

在这里，景观可能由许多海洋组成，每个海洋都无限大。

In this scenario, large pools that are not connected to the infinite seas become exceedingly rare.

在这种情况下，不与无边大海相连的大型水池就变得极其罕见。

That’s because a large, isolated pool can only stay separate if there is a lot of dry land — or closed edges — around it.

这是因为，一个大型的、孤立的水池只有在周围有大量干燥的土地或封闭的边缘时才能保持独立。

But a proof of supercritical sharpness seemed unattainable.

但超临界清晰度的证明似乎无法实现。

For one thing, the previous work was no help: A proof of supercritical sharpness on lattices was long and complicated, and it couldn’t be adapted to the more general case.

一方面，之前的工作没有帮助：晶格上超临界锐度的证明又长又复杂，而且无法适应更一般的情况。

While other foundational results were simplified in the last decade, “this was the one remaining fortress,” Nachmias said.

虽然其他基础性结果在过去十年中得到了简化，但“这是仅存的一个堡垒，”纳赫米亚斯说。

Mathematicians working on this problem “did some very beautiful things, initiated the theory, picked all the low-hanging fruit,” Benjamini said.

本杰明说，研究这个问题的数学家“做了一些非常漂亮的事情，开创了理论，摘取了所有容易实现的成果”。

“And then we started hitting the wall.”

“然后我们就开始碰壁了。”

In 2008, as progress on non-lattice percolation slowed, Schramm died at age 46 in a fall while hiking.

2008 年，随着非晶格渗透研究进展放缓，施拉姆在徒步旅行时摔倒身亡，享年 46 岁。

“We lost a genius, Oded Schramm, to a tragic accident,” Benjamini said.

“我们因一场悲惨事故失去了一位天才，奥德·施拉姆，”本杰明尼说。

“And then we needed to wait for some new geniuses to come.”

“然后我们需要等待一些新天才的到来。”

About a decade ago, the field began to accelerate again.

大约十年前，该领域再次开始加速发展。

But proving supercritical sharpness remained difficult.

但证明超临界清晰度仍然很困难。

Then, the team in Zurich produced a simple proof.

然后，苏黎世的团队给出了一个简单的证明。

Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn’t intend to prove supercritical sharpness.

迪斯金、埃索、拉达克里希南、苏达科夫和塔西翁并不打算证明超临界的敏锐度。

For most of fall 2025, they were trying to understand how critical probability scales with the number of edges in graphs.

在 2025 年秋季的大部分时间里，他们试图了解临界概率如何随图中边的数量而变化。

But the five mathematicians wanted results by the end of the semester.

但这五位数学家希望在学期末得到结果。

As that deadline neared, they still had nothing resembling a proof.

随着截止日期的临近，他们仍然没有任何类似的证据。

So Easo suggested pivoting to sharpness.

因此，伊索建议转向清晰度。

He, Diskin, and Radhakrishnan made some progress and brought their results to Sudakov and Tassion.

他、迪斯金和拉达克里希南取得了一些进展，并将他们的成果带给了苏达科夫和塔松。

As Tassion took in their work, an idea — perhaps an outrageous one — formed in his mind.

当塔松欣赏他们的作品时，一个想法——也许是一个令人震惊的想法——在他的脑海中形成。

He thought that, with some tweaks, their strategy might be strong enough to prove sharpness for all infinite transitive graphs.

他认为，通过一些调整，他们的策略可能足够强大，足以证明所有无限传递图的清晰度。

“From there, it was in my head day and night,” Tassion said.

“从那时起，它日夜萦绕在我的脑海里，”塔森说。

“Vincent went crazy with it,” Diskin said.

“文森特对此感到疯狂，”迪斯金说。

“I think he didn’t sleep for two weeks at least.”

“我想他至少有两周没有睡觉了。”

It wasn’t only Tassion.

不仅仅是塔西翁。

Over those weeks, the collaboration became frenzied.

在那几周里，合作变得疯狂。

The mathematicians traded ideas constantly, often texting late at night.

数学家们经常交换想法，经常在深夜发短信。

“We really all had this hunch that there might be something to it,” Diskin said.

迪斯金说：“我们确实都有这样的预感，认为其中可能有什么道理。”

“We were half joking at the beginning …

“一开始我们是半开玩笑的……

maybe the same idea could resolve this huge conjecture.

也许同样的想法可以解决这个巨大的猜想。

We were all laughing at each other, but what if, what if?”

我们都在互相嘲笑，但是万一呢，万一呢？”

Brimming with excitement, and with the holidays looming, they decided it was time to get serious and write their paper.

带着兴奋的心情，随着假期的临近，他们决定是时候认真写论文了。

To prove that a large isolated pool of fluid is unlikely above the critical probability, the mathematicians assumed they had such a pool and studied the surrounding shoreline.

为了证明一个巨大的孤立流体池不可能高于临界概率，数学家假设他们有这样一个池，并研究了周围的海岸线。

Along that shoreline, there were streams emptying into the pool, but there were also streams that linked back to one of the infinite seas.

沿着那条海岸线，有溪流流入水池，但也有溪流汇回一望无际的大海。

If those streams coincided anywhere, the mathematicians would have a contradiction — their so-called finite pool of fluid would actually be part of an infinite sea.

如果这些溪流在任何地方重合，数学家就会遇到矛盾——他们所谓的有限流体池实际上是无限海洋的一部分。

Maybe the same idea could resolve this huge conjecture.

也许同样的想法可以解决这个巨大的猜想。

We were all laughing at each other, but what if, what if?

我们都在互相嘲笑，但是万一呢，万一呢？

If the pool was big, the shoreline was long — meaning a larger area where the pool might connect to one of the infinite seas.

如果泳池很大，海岸线就会很长，这意味着泳池可以连接到一望无际的大海的区域更大。

The fivesome showed that this made it nearly impossible to avoid the contradiction.

五人组表明，这使得矛盾几乎无法避免。

As they hammered out the last details of their paper, they suddenly saw that with a simple change, their argument could be drastically improved.

当他们敲定论文的最后细节时，他们突然发现，通过一个简单的改变，他们的论点可以得到极大的改进。

They had been using a common technique in probability theory called sprinkling: They set aside a few of their open edges, corresponding to a slight lowering of the critical probability.

他们一直在使用概率论中的一种常见技术，称为洒水：他们留出一些开放边缘，对应于临界概率的轻微降低。

They then looked for a large pool among the rest of the edges and analyzed the open paths around it.

然后，他们在其余边缘中寻找一个大水池，并分析了它周围的开放路径。

Since the set-aside edges had nothing to do with the pool, they could be analyzed independently.

由于预留的边缘与水池无关，因此可以独立分析它们。

That made it easier to prove that, once combined with the rest of the graph, they almost always created a path to one of the infinite seas.

这使得更容易证明，一旦与图表的其余部分结合起来，它们几乎总是创建一条通往无限海洋之一的路径。

But as they talked, they hit upon an unorthodox improvement to this strategy.

但当他们交谈时，他们突然想到了对这一策略的非正统改进。

If they analyzed the sprinkles first, the proof got a lot simpler.

如果他们首先分析洒水，证明就会简单得多。

What’s more, it strengthened the argument enough that it worked for all infinite transitive graphs.

更重要的是，它充分强化了论证，使其适用于所有无限传递图。

“We had this ping-pong of ideas,” Diskin said.

“我们有各种各样的想法，”迪斯金说。

“Every time you throw ideas one at another, suddenly this wall becomes more blurry, until it vanishes completely.

“每当你互相抛出想法时，这堵墙就会突然变得更加模糊，直到完全消失。

Then it’s a bit scary, because you might actually have it.”

然后就有点害怕了，因为你可能真的拥有它。”

Finally, they were sure they had proved it: If the probability is anywhere above the critical threshold, even just a smidge, then fluid covers nearly the entire transitive graph.

最后，他们确信自己已经证明了这一点：如果概率高于临界阈值，即使只是一点点，那么流体几乎覆盖整个传递图。

Two months later, they posted a paper .

两个月后，他们发表了一篇论文。

Their argument applies to percolation on any infinite transitive graph.

他们的论点适用于任何无限传递图上的渗透。

“If you zoom into every sentence in the proof, it feels very familiar and simple, but the way they put it all together is genuinely novel,” Nachmias said.

“如果你放大证明中的每一个句子，你会感觉非常熟悉和简单，但他们将它们组合在一起的方式确实很新颖，”纳赫米亚斯说。

There is no shortage of unstudied percolation systems that their technique could apply to — like graphs where the nodes don’t all look identical, or more complicated models that describe freezing water or quantum materials.

他们的技术可以应用于许多未经研究的渗透系统，比如节点看起来并不完全相同的图表，或者描述冷冻水或量子材料的更复杂的模型。

A major question remains, though: On three-dimensional lattices — the graphs that most closely mirror physical systems — what happens exactly at the critical probability?

然而，一个主要问题仍然存在：在三维晶格（最接近地反映物理系统的图形）上，在临界概率下到底会发生什么？

Is there an infinite sea?

世界上有无边无际的大海吗？

The progress on the problem is especially significant to Benjamini, who waited a decade for his expedition to start up again.

这个问题的进展对本杰明尼来说尤其重要，他等待了十年才再次开始他的探险活动。

“For the community, for us, it’s a very deep and meaningful theorem, and it’s a part of the puzzle,” he said.

“对于社区，对于我们来说，这是一个非常深刻且有意义的定理，也是谜题的一部分，”他说。

Of the proof, Benjamini said, “it’s a gem.

对于这个证据，本杰明尼说：“这是一颗宝石。

It’s a gem.”

这是一颗宝石。”

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辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

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