内容来源：https://www.quantamagazine.org/stunning-percolation-proof-solves-decades-old-puzzle-about-phase-transitions-20260828/

内容总结：

渗透理论重大突破：五位数学家攻克数十年未解之谜

近日，来自苏黎世联邦理工学院的五位数学家成功解决了一个困扰学界数十年的渗透理论核心难题，该成果被同行评价为“令人惊叹的证明”。

圣诞前夕的突破

2025年圣诞节前一周，五位数学家——博士后Sahar Diskin、Philip Easo，研究生Ritvik Ramanan Radhakrishnan，以及教授Benny Sudakov和Vincent Tassion——在苏黎世联邦理工学院的教室里紧张工作。他们即将迎来职业生涯中具有里程碑意义的突破。

研究团队正在攻克渗透理论中最大的开放性问题之一。渗透理论研究的是网络中的流动现象，其典型例子包括热水渗透咖啡粉床。他们试图解决关于图（由点和线组成的网络）如何被大型连通区域（相当于流体汇聚的“水潭”）覆盖的基本问题。

“一开始我们几乎不敢相信，”Radhakrishnan回忆道。他们争分夺秒地验证每一个细节，甚至顾不上圣诞假期。“我不确定我们的女友和家人是否像我们一样开心。但那一刻我们都非常兴奋，”Diskin说，“能遇到如此重大而有意义的发现，实在难得。”

他们通宵达旦工作，到12月17日清晨，确信自己的想法是正确的。到圣诞节时，他们完成了一个完整的证明。

从煤炭到数学模型

渗透理论可以描述多种流动现象：病毒在城市中的传播、气体通过过滤器、野火的蔓延。但它的起源灵感来自煤炭。

20世纪40年代，后来因DNA结构研究闻名的科学家罗莎琳德·富兰克林在英国煤炭利用研究协会工作，试图理解煤炭、木炭和石墨的复杂性质。科学家们知道煤炭布满微小孔洞，但不清楚为何有些煤能让流体通过，而另一些则不能。

大约十年后，研究人员Simon Broadbent和John Hammersley为理解防毒面具中的碳过滤器，开发了一个数学模型。其思路很简单：在均匀分布的点阵（晶格）中，对每对相邻点抛硬币决定是否连接。当连接概率超过一个称为“临界概率”的阈值时，晶格突然“打开”，流体可以广泛流动。

关键猜想与长期困扰

研究人员数十年来一直试图量化渗透相变过程，预测流体“水潭”的增长速度。许多学者预测水潭增长非常迅速——低于临界概率时，水潭极小；高于临界概率时，单个“海洋”几乎覆盖一切。这一预测被称为“尖锐性猜想”。

20世纪80年代，尖锐性猜想在晶格上得到证明，成为该领域的奠基性成果。1996年，Itai Benjamini和Oded Schramm开始将渗透理论扩展到更广泛的“传递图”类别。他们怀疑尖锐性猜想在所有无限传递图上都成立，但这猜想分为两个部分：“次临界”部分（临界点以下的情况）于2007年由Tonći Antunović和Ivan Veselić完成，但“超临界”部分（临界点以上的情况）的证明似乎遥不可及。

特拉维夫大学的Asaf Nachmias表示，此前在晶格上的超临界尖锐性证明冗长复杂，无法推广到更一般的情况。尽管过去十年其他基础性成果不断简化，“但这是最后一座堡垒”。

2008年，正当非晶格渗透研究进展放缓时，Oded Schramm在徒步中意外去世，年仅46岁。Benjamini悲痛地说：“我们因一场悲剧事故失去了一位天才。然后我们需要等待新的天才出现。”

化繁为简的巧妙证明

Diskin、Easo、Radhakrishnan、Sudakov和Tassion最初并未打算证明超临界尖锐性。2025年秋季的大部分时间，他们在研究临界概率如何随图的边数变化。但随着学期结束临近，他们仍没有取得实质性进展。Easo建议转向尖锐性问题。

当他们取得一些进展后，Tassion产生了灵感——他意识到，经过一些调整，他们的策略可能足以证明所有无限传递图的尖锐性。“从那以后，我日日夜夜都在想这个问题，”Tassion说。“Vincent为此疯狂了，”Diskin笑道，“我觉得他至少有两周没睡过觉。”

他们使用了概率论中常用的“喷洒”技术，但进行了非传统的改进——先分析“喷洒”的边，使证明大大简化，且足以适用于所有无限传递图。“我们就像在打乒乓球，”Diskin说，“每当你把想法相互碰撞，这堵墙就变得更加模糊，直到完全消失。然后你会感到有点害怕，因为你可能真的做到了。”

两个月后，他们发布了论文。该证明适用于任何无限传递图上的渗透现象。Nachmias评价道：“如果逐句审视这个证明，每个部分都感觉很熟悉、很简单，但他们的组合方式是真正新颖的。”

深远意义

这一突破对等待了十年的Benjamini尤其重要。“对学界、对我们来说，这是一个非常深刻而有意义的定理，是整个拼图的一部分。”他评价道，“这是一颗宝石，一颗真正的宝石。”

目前仍有一个重要问题悬而未决：在三维晶格上（最接近物理系统的图），在临界概率处究竟会发生什么？是否存在无限大的“海洋”？但无论如何，这项成果为渗透理论研究开辟了新的道路，其方法还可应用于其他类型的图和更复杂的物理模型。

中文翻译：

“惊艳”的渗流证明解开相变难题，悬而未决数十年的谜题终获解答

引言

2025年圣诞节前一周，五位数学家驻扎在苏黎世联邦理工学院的一间教室里。气氛热烈而紧张：他们距离一项足以定义职业生涯的重大突破仅一步之遥。

这个团队——由当时的博士后萨哈尔·迪斯金和菲利普·伊索，研究生里特维克·拉马南·拉达克里希南，本尼·苏达科夫以及文森特·塔西翁组成——正在完善渗流理论（研究网络中流动的学科）中最大开放问题之一的解决方案。

渗流涵盖了极为广泛的现象，但其典型例子涉及流体，比如热水渗透过咖啡粉层。迪斯金、伊索、拉达克里希南、苏达科夫和塔西翁试图解决一个关于图（由线或边连接的点构成的网络）如何被大型连通区域（相当于流体汇聚的“水潭”）所占据的根本性问题。这个团队瞥见了一个简洁的论证思路，可以同时处理种类繁多的图。

“我们一开始几乎不敢相信，”拉达克里希南说。

他们争分夺秒地核实每一个细节，急于在灵感消散之前将想法固定下来，完全顾不上假期。“我不确定女朋友和家人是否像我们一样开心。但那一刻我们所有人都非常兴奋，”迪斯金说。“能碰到如此宏大、如此有意义的东西，真的非常罕见。”

他们通宵工作。到12月17日早晨，在极度兴奋的状态下，他们确信自己的想法是正确的。到圣诞节时，他们已经完成了完整的证明。

他们回答了一个困扰数十年的问题：当一个渗流网络向流体开放时，淹没的速度究竟有多快。“这个证明让我感到极大的喜悦，”特拉维夫大学研究渗流理论和概率论的阿萨夫·纳赫米亚斯说。“它令人惊艳。”

本杰明·苏达科夫

富兰克林的流体

渗流可以描述多种流动：病毒在城市中的传播、气体穿过过滤器、野火的蔓延。但它最初灵感来自煤炭。

20世纪40年代，科学家罗莎琳·富兰克林——如今因在DNA结构方面的工作而闻名——受雇于英国煤炭利用研究协会，试图理解煤炭、木炭和石墨的复杂性质。科学家们知道煤炭中布满微小孔隙，但他们不知道为何某些类型的煤允许流体通过，而另一些则不可渗透。

通过将煤炭浸入各种流体中，富兰克林得以测量其孔隙的典型尺寸以及变异程度。大约十年后，研究人员西蒙·布罗德本特和约翰·哈默斯利——为了理解防毒面具中的碳过滤器——发展出一个数学模型。

他们的想法很简单。取一个均匀分布点的网格（也称为格点）和一枚硬币。对于每对相邻点，掷一次硬币。如果正面朝上，就用一条边连接这两个点，流体可以在这些点之间流动。如果背面朝上，则流动被阻断。对每一对点重复这个过程。流体能走多远？

马克·贝兰/《量子》杂志

答案取决于硬币正面朝上的概率，这个概率可以从0%到100%变化。当概率较低时，流体只能通过少数通道流动，因此会在小而孤立的水洼中聚集。

但一旦概率超过一个称为临界概率的阈值，格点突然打开。流体可以广泛地穿过整个系统。

临界概率的确切值随格点的形状而变化——正方形格点的临界概率不同于三角形格点，三维格点的临界概率不同于二维格点。但当你越过临界值之上时，你会看到一个相变，就像液态水变成冰一样。在有限图上，跨越临界概率意味着网络将被一个巨大的流体海洋所主导。在无限图上——一种图在所有方向上无限延伸的抽象——一个或几个无限的海洋将占据主导。物理学家很快意识到，通过渗流，他们可以研究融化和冻结，以及其他相变，如磁化。

“物理学中的相变很难进行严格的数学研究，”伊索说。“渗流就像是它的卡通简化版。所以人们往往先研究这个，然后工具再慢慢渗透下去。”

对于那些长期被复杂现实世界相变困扰的科学家来说，“这是在非常简单的模型中抓住了本质，”魏茨曼科学研究所的伊塔伊·本杰米尼说。“许多可能使问题模糊化的因素都被剥离了。”

伊塔伊·本杰米尼供图

几十年来，研究人员致力于量化渗流相变。他们想知道，随着硬币正面朝上的概率增加，流体水潭究竟能以多快的速度增长。许多人预测水潭增长得非常快——低于临界概率时，水洼很小；高于临界概率时，一个单一的海洋几乎覆盖一切。这个预测被称为“尖锐性猜想”。

当尖锐性在20世纪80年代在格点上被证明时——由两个独立团队完成，一个在新泽西，一个在莫斯科——它是“奠基性的，”普林斯顿大学和加州理工学院的汤姆·哈奇克罗夫特说，他是伊索的博士生导师。知道了尖锐性，数学家可以推断出网络中被淹没部分的大量结构信息——特别是，它看起来与底层格点非常相似。

因此，当本杰米尼和他的同事奥代德·施拉姆计划将渗流拓展到新型网络时，自然会产生疑问：尖锐性是否也会随之而来。

脱离网格

1996年，本杰米尼和施拉姆想研究一类更大的图中的渗流，这类图称为传递图。要理解什么是传递图，可以把图想象成平坦荒凉的景观上的道路网络。如果想知道自己在这些路上的位置，唯一的地标是交叉路口。但如果图是传递的，所有交叉路口看起来都一样——要弄清自己在哪里，就需要GPS或指南针。

正方形格点是传递图的一个例子：每个交叉路口由四条边以直角相交组成。但传递图有很多种——简单的环（下图左）和无限扩展的“树”（下图右），以及一些几乎无法可视化的图。

许多传递图代表了其他数学子领域（如代数或几何）中的对象。本杰米尼对这些跨学科的可能性很感兴趣——他希望渗流过程能揭示图本身的一些洞见。“你有一个舞台，即几何，还有一个舞者，即随机过程，”他说。通过观察舞者，他希望更多地了解舞台。

在接下来的十年里，本杰米尼、施拉姆及其同事发表了一系列关于传递图渗流的结果。他们证明，对于一类无限传递图，当你打开边让流体流动时，渗流表现出相变：小而孤立的流体水潭突然凝聚成一个由连通的河流组成的无限网络。

但他们仍然不知道这个转变发生得有多快。在临界点以下，水潭有多大、有多少？在临界点以上，那个无限网络是一片溪流纵横的草地——还是更像一个海洋，淹没了整个图？

本杰米尼和施拉姆怀疑尖锐性猜想的一个版本在所有无限传递图上都成立。这个猜想可以分解为两个独立的问题。“次临界”部分——处理临界点以下发生的情况——由通契·安图诺维奇和伊万·韦塞利奇于2007年完成。他们的工作表明，在此区域，流体水潭又小又分散。即使在临界点以下仅一丝一毫，系统看起来也更像亚利桑那州而非明尼苏达州。

猜想的“超临界”部分——处理临界阈值以上的概率——看起来更难。在这种情景下，景观应该由可能多个海洋组成，每个海洋都是无限大的。在这种情形下，不与无限海洋连接的大型水潭变得极为罕见。这是因为一个大型孤立水潭只有在周围有大量干燥土地（即关闭的边）时才能保持独立。

但超临界尖锐性的证明似乎难以企及。一方面，之前的工作帮不上忙：格点上超临界尖锐性的证明冗长复杂，无法推广到更一般的情形。虽然其他基础性结果在过去十年中被简化了，“但这是唯一剩下的堡垒，”纳赫米亚斯说。

致力于这个问题的数学家“做了非常漂亮的工作，开创了这一理论，摘取了所有低垂的果实，”本杰米尼说。“然后我们开始撞墙。”

2008年，当非格点渗流的进展放缓时，施拉姆在徒步时意外坠落身亡，年仅46岁。“我们失去了一位天才，奥代德·施拉姆，一场悲惨的事故，”本杰米尼说。“然后我们需要等待新的天才出现。”

大约十年前，这个领域开始再次加速。但证明超临界尖锐性仍然困难。

然后，苏黎世的团队给出了一个简洁的证明。

一个急转弯

迪斯金、伊索、拉达克里希南、苏达科夫和塔西翁并没有打算证明超临界尖锐性。在2025年秋季的大部分时间里，他们试图理解临界概率如何随图中的边数变化。

但这五位数学家希望在学期结束前取得成果。随着截止日期临近，他们仍然没有任何接近证明的东西。于是伊索建议转向尖锐性。他、迪斯金和拉达克里希南取得了一些进展，并把结果带给苏达科夫和塔西翁。当塔西翁消化他们的工作时，一个想法——也许是一个大胆的想法——在他脑海中形成。

他认为，经过一些调整，他们的策略可能有足够的力量为所有无限传递图证明尖锐性。“从那以后，我日日夜夜都在想这件事，”塔西翁说。

“文森特为此疯狂了，”迪斯金说。“我觉得他至少两周没睡觉。”

不只是塔西翁。在那几周里，合作变得狂热。数学家们不断交流想法，经常深夜发消息。“我们真的都有这种预感，这件事可能有点名堂，”迪斯金说。“一开始我们半开玩笑……也许同一个想法能解决这个巨大的猜想。我们都在互相笑话，但万一呢，万一呢？”

极致的简洁

满怀兴奋，假期将至，他们决定是时候认真起来，开始撰写论文了。

为了证明在临界概率以上，大型孤立流体水潭不太可能出现，数学家们假设存在这样一个水潭，并研究其周围的岸线。沿着岸线，有溪流汇入水潭，但也有溪流连接回某个无限海洋。如果这些溪流在任何地方交汇，数学家就会得到一个矛盾——他们所谓的有限流体水潭实际上会是无限海洋的一部分。

如果水潭很大，岸线就很长——意味着更大的区域可能让水潭与某个无限海洋连接。五人团队证明了这使得矛盾几乎不可避免。

当他们敲定论文的最后一个细节时，他们突然发现，只需一个简单的改变，他们的论证就能得到大幅改进。

他们一直在使用概率论中一种常见的技术，称为“喷洒法”：他们预留一些开放的边，相当于稍微降低临界概率。然后他们在其余边中寻找一个大水潭，并分析其周围的开放路径。由于预留的边与水潭无关，它们可以独立分析。这使得证明更容易：一旦与图的其余部分结合，它们几乎总能创建一条通向某个无限海洋的路径。

但在讨论中，他们想到了对这一策略的非正统改进。如果先分析“喷洒”的边，证明会大大简化。更重要的是，这加强了论证，使其对所有无限传递图都成立。“我们进行着这种想法的乒乓对打，”迪斯金说。“每次你把想法互相抛出，突然那堵墙就变得更模糊，直到完全消失。然后有点可怕，因为你可能真的成功了。”

最终，他们确信自己已经证明：如果概率在临界阈值以上的任何位置，哪怕只高出一丁点，那么流体就几乎覆盖了整个传递图。

两个月后，他们发布了一篇论文。他们的论证适用于任何无限传递图上的渗流。“如果你放大证明中的每一句话，感觉非常熟悉和简单，但他们将其组合在一起的方式是真正新颖的，”纳赫米亚斯说。

他们的技术可以应用于大量尚未研究的渗流系统——比如节点并非全部相同的图，或者描述冻结水或量子材料的更复杂模型。

不过仍然存在一个重大问题：在三维格点上——最接近物理系统的图——在临界概率处究竟会发生什么？是否存在一个无限海洋？

在这个问题上取得的进展对本杰米尼尤其意义重大，他等待了十年才看到探索重新启动。“对于这个领域，对于我们来说，这是一个非常深刻且有意义的定理，它是拼图的一部分，”他说。

谈到这个证明，本杰米尼说：“它是一颗宝石。它是一颗宝石。”

英文来源：

‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions
Introduction
The week before Christmas 2025, five mathematicians were holed up in a classroom at ETH Zurich. The mood was electric: They were this close to a career-defining breakthrough.
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.
Percolation captures a vast array of phenomena, but the prototypical examples involve fluids, like hot water seeping through a bed of coffee grounds. 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. The group had glimpsed a simple argument that could deal with a huge variety of graphs at once.
“We almost didn’t believe it at first,” Radhakrishnan said.
They raced to confirm each detail, eager to get their idea down before it shimmered away — and heedless of the holiday. “I’m not sure the girlfriends and the families were as happy as we were. But we were all very happy at that moment,” Diskin said. “It’s really rare that you’re able to hit something that feels so big and so meaningful.”
They worked through the night. By the morning of December 17, exhilarated from the effort, they were convinced their idea was correct. By Christmas, they’d nailed down a proof.
They had answered a decades-old question about how fast a percolation network floods as you open it up to fluid flow. “I find great joy in this proof,” said Asaf Nachmias of Tel Aviv University, who studies percolation theory and probability. “It’s stunning.”
Benjamin Sudakov
Franklin’s Fluids
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. But its original inspiration was coal.
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. 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.
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. About a decade later, the researchers Simon Broadbent and John Hammersley — wanting to understand the carbon filters in gas masks — developed a mathematical model.
Their idea was simple. Take a grid of evenly spaced points (also called a lattice) and a coin. For each pair of neighboring points, flip the coin. If it lands on heads, connect the points with an edge. Fluid can flow between these points. If the coin lands on tails, the flow is blocked. Repeat this procedure for every pair of points. How far does the fluid go?
Mark Belan/Quanta Magazine
The answer depends on the probability that your coin lands on heads, which can range from 0% to 100%. When the probability is low, fluid can flow through only a few channels, and so it collects in small, isolated puddles.
But once the probability passes a threshold called the critical probability, the lattice suddenly opens up. Fluid can travel extensively through the system.
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. But as you move above that critical value, you’ll see a phase transition, like liquid water turning to ice. On a finite graph, crossing the critical probability means the network will become dominated by one large sea of fluid. On an infinite graph — an abstraction where the graph extends forever in all directions — one or several infinite seas will dominate. Physicists quickly realized that through percolation, they could learn about melting and freezing, as well as other phase transitions like magnetization.
“Phase transitions in physics are very hard to study rigorously,” Easo said. “Percolation is like the caricature. So people often try to study that first, and then tools trickle down.”
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. “A lot of things that could cloud the issue were removed.”
Courtesy of Itai Benjamini
For decades, researchers worked to quantify the percolation phase transition. They wanted to know exactly how quickly pools of fluid can grow as you increase the probability that your coin lands on heads. 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. This prediction is called the sharpness conjecture.
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. 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.
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.
Off the Grid
In 1996, Benjamini and Schramm wanted to study percolation in a much larger class of graphs, called transitive graphs. To understand what a transitive graph is, imagine the graph as a network of roads on a flat, desolate landscape. If you want to know where you are on these roads, the only landmarks are the intersections. But if the graph is transitive, all the intersections look similar — to figure out where you are, you’ll need GPS or a compass.
A square lattice is one example of a transitive graph: Every intersection consists of four edges meeting at right angles. 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.
Many transitive graphs represent objects from other mathematical subfields, like algebra or geometry. Benjamini was intrigued by these interdisciplinary possibilities — he hoped the percolation process would reveal insights into the graph itself. “You have a stage, which is geometry, and a dancer, which is the random process,” he said. By watching the dancer, he hoped to learn more about the stage.
Over the next decade, Benjamini, Schramm, and their colleagues published a flurry of results on the percolation of transitive graphs. 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.
But they still didn’t know how fast that transition happened. Below the critical point, how big and how numerous were the pools? Above it, was the infinite web a meadow crisscrossed with streams — or was it more like an ocean, swamping the entire graph?
Benjamini and Schramm suspected that a version of the sharpness conjecture was true on all infinite transitive graphs. That conjecture could be broken down into two separate problems. The “subcritical” half — addressing what happens below the critical point — was completed in 2007, by Tonći Antunović and Ivan Veselić. Their work showed that here, pools of fluid are tiny and far apart. Even a hair below the critical point, the system looks more like Arizona than Minnesota.
The “supercritical” half of the conjecture — which deals with probabilities above the critical threshold — seemed harder. Here, the landscape should be made up of possibly many seas, each infinitely large. In this scenario, large pools that are not connected to the infinite seas become exceedingly rare. That’s because a large, isolated pool can only stay separate if there is a lot of dry land — or closed edges — around it.
But a proof of supercritical sharpness seemed unattainable. 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. While other foundational results were simplified in the last decade, “this was the one remaining fortress,” Nachmias said.
Mathematicians working on this problem “did some very beautiful things, initiated the theory, picked all the low-hanging fruit,” Benjamini said. “And then we started hitting the wall.”
In 2008, as progress on non-lattice percolation slowed, Schramm died at age 46 in a fall while hiking. “We lost a genius, Oded Schramm, to a tragic accident,” Benjamini said. “And then we needed to wait for some new geniuses to come.”
About a decade ago, the field began to accelerate again. But proving supercritical sharpness remained difficult.
Then, the team in Zurich produced a simple proof.
A Sharp Turn
Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn’t intend to prove supercritical sharpness. For most of fall 2025, they were trying to understand how critical probability scales with the number of edges in graphs.
But the five mathematicians wanted results by the end of the semester. As that deadline neared, they still had nothing resembling a proof. So Easo suggested pivoting to sharpness. He, Diskin, and Radhakrishnan made some progress and brought their results to Sudakov and Tassion. As Tassion took in their work, an idea — perhaps an outrageous one — formed in his mind.
He thought that, with some tweaks, their strategy might be strong enough to prove sharpness for all infinite transitive graphs. “From there, it was in my head day and night,” Tassion said.
“Vincent went crazy with it,” Diskin said. “I think he didn’t sleep for two weeks at least.”
It wasn’t only Tassion. Over those weeks, the collaboration became frenzied. The mathematicians traded ideas constantly, often texting late at night. “We really all had this hunch that there might be something to it,” Diskin said. “We were half joking at the beginning … maybe the same idea could resolve this huge conjecture. We were all laughing at each other, but what if, what if?”
Radical Simplicity
Brimming with excitement, and with the holidays looming, they decided it was time to get serious and write their paper.
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. Along that shoreline, there were streams emptying into the pool, but there were also streams that linked back to one of the infinite seas. 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.
If the pool was big, the shoreline was long — meaning a larger area where the pool might connect to one of the infinite seas. The fivesome showed that this made it nearly impossible to avoid the contradiction.
As they hammered out the last details of their paper, they suddenly saw that with a simple change, their argument could be drastically improved.
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. They then looked for a large pool among the rest of the edges and analyzed the open paths around it. Since the set-aside edges had nothing to do with the pool, they could be analyzed independently. 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.
But as they talked, they hit upon an unorthodox improvement to this strategy. If they analyzed the sprinkles first, the proof got a lot simpler. What’s more, it strengthened the argument enough that it worked for all infinite transitive graphs. “We had this ping-pong of ideas,” Diskin said. “Every time you throw ideas one at another, suddenly this wall becomes more blurry, until it vanishes completely. Then it’s a bit scary, because you might actually have it.”
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.
Two months later, they posted a paper. Their argument applies to percolation on any infinite transitive graph. “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.
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.
A major question remains, though: On three-dimensional lattices — the graphs that most closely mirror physical systems — what happens exactly at the critical probability? Is there an infinite sea?
The progress on the problem is especially significant to Benjamini, who waited a decade for his expedition to start up again. “For the community, for us, it’s a very deep and meaningful theorem, and it’s a part of the puzzle,” he said.
Of the proof, Benjamini said, “it’s a gem. It’s a gem.”