## From The California Institute of Technology: “Mathematically Percolating”

From The California Institute of Technology

8.5.22

Tom Hutchcroft explains why phase transitions in percolation models are so fascinating. Credit: Caltech.

These images show three different phases in a percolation model: before, at, and beyond the critical phase transition. As the probability of an edge being blue, or p, goes up (from left to right), it becomes easier for the blue portions to spread across the grid. At the point of the phase transition (middle), fractal-like clusters emerge that take long, meandering paths across the grid. Credit: Nils Berglund.

**When water flows through a bed of ground espresso beans, ultimately resulting in a delicious latte, the water is undergoing a process called percolation. The water slowly meanders through the coffee at just the right rate to extract the rich coffee flavors. In general, percolation refers to liquids filtering through a porous medium. The process can describe not only the generation of lattes, but also a host of other phenomena, such as how diseases spread and even physics concepts such as magnetism.**

For mathematicians like Tom Hutchcroft, who joined the Caltech faculty last year as a professor of mathematics, the most interesting aspect of percolation is what happens during a phase transition, the point where an abrupt qualitive change in the system occurs. “You only change one factor in the system a tiny bit, and then you get a big change,” he says. A classical phase transition occurs when water freezes.

In mathematical percolation models, phase transitions can result in “really interesting mathematical behavior,” according to Hutchcroft, including fractal patterns; fractal refers to self-similar patterns seen at different scales.

Hutchcroft is studying how the geometry of fractal trees like the one depicted here change when viewed in different dimensions.

Hutchcroft, who was born and raised in England, earned his bachelor’s degree in mathematics from Cambridge University in 2013 and his PhD in mathematics from the University of British Columbia, Canada, in 2017. He held internships at Microsoft Research Theory Group during his graduate studies, and later completed postdoctoral fellowships at University of Cambridge from 2017 to 2021.

We met with Hutchcroft over Zoom to learn more about the math of percolation and what he is enjoying about Caltech so far.

**What does percolation have to do with the spread of a disease?**

Percolation theory is a way of describing clustered components in random networks and can be applied to complex things like the spread of an infection through a population. Systems like these have phase transitions. With epidemics, there’s a critical point or phase transition called “R nought,” or R0. This value depends on the average number of people that an infected person infects. When R nought is below 1, the epidemic will die out; when it’s above 1, it will grow exponentially. When R is exactly 1, the infection will die out but very slowly. When you look at this point in models, you tend to have a lot of mathematically interesting behavior. And when you use branching, tree-like models for epidemics, which are a form of a percolation model, you’ll get some interesting fractal geometry in the tree. This has been well understood since the 1990s. But even though you are doing something very simple, you get really mathematically rich objects coming out at the end.

**How do mathematicians study these percolation models?**

While physicists and other scientists may study the statistical physics or statistical mechanics of similar systems as a means to explain the behavior of the components, we mathematicians are interested in the pure math, which can be very complex and interesting. In general, we draw out grids, with edges and nodes, where the edges connect the nodes. These are the percolation models that explain how liquid can flow through a porous media. Imagine that for each edge of this grid, you flip a coin that has a probability “p” of being heads. If the coin comes out heads, you keep the edge, and if it comes out tails, you delete the edge.

When p is small, or the probability of keeping an edge is small, you will end up with small clusters of connections that are like small islands that don’t connect to anything else. When p is greater than the critical parameter at which a phase transition occurs, called pc [pronounced pee-cee], you will get one big, connected cluster. When p is exactly equal to pc, we expect to get large, fractal-like clusters that permeate across the grid but do so in a zero-density way, with extremely long, tortuous paths.

So, if this model were explaining coffee percolation, then when p is less than pc, the water would not get through the coffee—it would get stuck in the islands of small clusters. When p is larger than pc, the water would readily flow through. When p is equal to pc, at the phase transition, the water would slowly meander through the grinds, which is what you would want for a good cup of espresso.

If you look at the connections between the nodes at this phase transition and under different scales you will start to see the fractal and winding math.

**What problems are you working on in this field?**

The two-dimensional models are very well understood and even models with 100 dimensions are easier to understand. But the three- four- and five-dimensional cases are extremely hard to study. One thing I’m working on is trying to crack the three-dimensional problem. The most basic question is to figure out if the phase transition occurs with a jump-what we call discontinuous-or more smoothy-what we call continuous. This problem has been open for a really long time and needs to be solved before we can move on to understanding all the cool fractal stuff that should be happening at the phase transition. I’m also working on other related problems, such as long-range percolation where the probability of two nodes having an edge between them depends on the distance between the nodes. Changing how this probability falls off with the distance has a surprisingly similar effect to changing the dimension of the grid and lets us treat the dimension like a continuous parameter.

**What do you love about working on these math problems?**

A lot of the appeal for me is that it’s fun. When you get a good problem, you get hooked on it. Math is like the king of all puzzle games, but it goes beyond puzzles in that you the solution is very insightful. You not only solve the problem, but you build new conceptual frameworks for understanding other math problems.

**How do you like Caltech so far?**

One of the things that drew me to Caltech was the small class size. It feels less like lecturing and more like doing a seminar where you get to interact with everyone individually. I like the tight community here. Of course, the small size of Caltech can mean less interaction with mathematical peers, but a lot of that is going to be balanced out by the fact that the American Institute of Mathematics is moving its headquarters to Caltech. That’s going to bring a lot more activity here, and people will be passing through regularly.

I also like the mountains in the Pasadena area, which we don’t have back home in the UK. We’ve been going out hiking. You can look at a mountain on campus and then get in the car and be there in 15 minutes.

**Anything else you’d like to add?**

Since coming to Caltech, I’ve also set up the LA Probability Forum, a monthly mini-conference for the LA-area probability community, so that I get to regularly interact with my colleagues at UCLA and USC, and anyone else who would like to be involved. This has been really enriching for me both scientifically and socially.

See the full article here .

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The The California Institute of Technology is a private research university in Pasadena, California. The university is known for its strength in science and engineering, and is one among a small group of institutes of technology in the United States which is primarily devoted to the instruction of pure and applied sciences.

The California Institute of Technology was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began attracting influential scientists such as George Ellery Hale, Arthur Amos Noyes, and Robert Andrews Millikan in the early 20th century. The vocational and preparatory schools were disbanded and spun off in 1910 and the college assumed its present name in 1920. In 1934, The California Institute of Technology was elected to the Association of American Universities, and the antecedents of National Aeronautics and Space Administration ‘s Jet Propulsion Laboratory, which The California Institute of Technology continues to manage and operate, were established between 1936 and 1943 under Theodore von Kármán.

The California Institute of Technology has six academic divisions with strong emphasis on science and engineering. Its 124-acre (50 ha) primary campus is located approximately 11 mi (18 km) northeast of downtown Los Angeles. First-year students are required to live on campus, and 95% of undergraduates remain in the on-campus House System at The California Institute of Technology. Although The California Institute of Technology has a strong tradition of practical jokes and pranks, student life is governed by an honor code which allows faculty to assign take-home examinations. The The California Institute of Technology Beavers compete in 13 intercollegiate sports in the NCAA Division III’s Southern California Intercollegiate Athletic Conference (SCIAC).

As of October 2020, there are 76 Nobel laureates who have been affiliated with The California Institute of Technology, including 40 alumni and faculty members (41 prizes, with chemist Linus Pauling being the only individual in history to win two unshared prizes). In addition, 4 Fields Medalists and 6 Turing Award winners have been affiliated with The California Institute of Technology. There are 8 Crafoord Laureates and 56 non-emeritus faculty members (as well as many emeritus faculty members) who have been elected to one of the United States National Academies. Four Chief Scientists of the U.S. Air Force and 71 have won the United States National Medal of Science or Technology. Numerous faculty members are associated with the Howard Hughes Medical Institute as well as National Aeronautics and Space Administration. According to a 2015 Pomona College study, The California Institute of Technology ranked number one in the U.S. for the percentage of its graduates who go on to earn a PhD.

**Research**

The California Institute of Technology is classified among “R1: Doctoral Universities – Very High Research Activity”. Caltech was elected to The Association of American Universities in 1934 and remains a research university with “very high” research activity, primarily in STEM fields. The largest federal agencies contributing to research are National Aeronautics and Space Administration; National Science Foundation; Department of Health and Human Services; Department of Defense, and Department of Energy.

In 2005, The California Institute of Technology had 739,000 square feet (68,700 m^2) dedicated to research: 330,000 square feet (30,700 m^2) to physical sciences, 163,000 square feet (15,100 m^2) to engineering, and 160,000 square feet (14,900 m^2) to biological sciences.

In addition to managing NASA-JPL/Caltech , The California Institute of Technology also operates the Caltech Palomar Observatory; the Owens Valley Radio Observatory;the Caltech Submillimeter Observatory; the W. M. Keck Observatory at the Mauna Kea Observatory; the Laser Interferometer Gravitational-Wave Observatory at Livingston, Louisiana and Hanford, Washington; and Kerckhoff Marine Laboratory in Corona del Mar, California. The Institute launched the Kavli Nanoscience Institute at The California Institute of Technology in 2006; the Keck Institute for Space Studies in 2008; and is also the current home for the Einstein Papers Project. The Spitzer Science Center, part of the Infrared Processing and Analysis Center located on The California Institute of Technology campus, is the data analysis and community support center for NASA’s Spitzer Infrared Space Telescope [no longer in service].

The California Institute of Technology partnered with University of California at Los Angeles to establish a Joint Center for Translational Medicine (UCLA-Caltech JCTM), which conducts experimental research into clinical applications, including the diagnosis and treatment of diseases such as cancer.

The California Institute of Technology operates several Total Carbon Column Observing Network stations as part of an international collaborative effort of measuring greenhouse gases globally. One station is on campus.

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