A Biography of Robert M Ziff

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Robert (Bob) Ziff was born in Los Angeles, California, and spent his early years there. In high school he was interested in math, physics, industrial drawing, ham radio, and building rockets and other projects with his older brother Stuart in their garage workshop. Stuart went on to do special effects in Hollywood (Star Wars, Ghostbusters, etc) and has a garage workshop to this day. His sister Janet is an artist working and living in Manhattan and Barcelona. His father was in the real estate business, after a period writing for radio and early TV, and his mother worked as a clerk in a VA hospital. During World War II, his father fought with Patton in Europe, and his mother worked in a plant in Santa Monica building Douglas aircraft. Stuart lives nearby in Santa Monica now.

At the age of 16, Bob entered UCLA physics, and enjoyed a chemistry class and project with chemist Willard F Libby (Nobel prize, 1960, for carbon-14 dating). He worked on projects studying liquid helium with physics professors Marvin Chester and Seth Putterman, a new faculty member at the time. With Chester, Bob published his first paper concerning dissipation in superfluid helium, now over 50 years old [1].

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Putterman recommended that Bob study at his alma mater Rockefeller University, in the heart of the Upper East side of Manhattan, and he went there and loved it after living in suburban Los Angeles, regularly frequenting Central Park, museums and Chinatown. At that time Rockefeller had an active group of physicists in statistical physics, including George Uhlenbeck (co-discoverer of the spin of the electron), E G D (Eddie) Cohen, and mathematician Mark Kac. He studied the Bose–Einstein transition of the ideal Bose gas, something Uhlenbeck himself explored when he was a graduate student fifty years earlier. He studied how the phase transition appears as the system size goes to infinity, and differences between the canonical and grand canonical ensembles, in particular for fluctuations. While at the time this subject was considered to be rather academic, in subsequent years the research became quite relevant as Bose–Einstein condensation in gases was produced in the laboratory, leading to several Nobel prizes. His PhD work on the Bose gas was published in its entirety as a Physics Report with Kac and Uhlenbeck [2]. After graduating from Rockefeller, Bob moved to Los Alamos National Laboratory as a post-doc with Laurence Campbell, again working on a problem related to liquid helium: the behavior of vortex arrays in rotating buckets. Because the superfluid component is irrotational, it can only go into rotation by creating an array of vortices which are quantized in strength (something independently postulated by Onsager and Feynman). The vortex arrays were imaged by Yarmchuk et al in 1979 [3], and soon thereafter Campbell and Ziff produced a catalog of the equilibrium rotating array patterns that have been used by many [4]. These same arrays have come up in other contexts, including in a rotating Bose–Einstein gas, which follows a similar theory. Bob also has many fond memories from living in New Mexico, and to this day enjoys the food (including roasted green chiles) from that enchanted land. After Los Alamos, he became a post-doc at Stony Brook with George Stell, whom he considers a great mentor and friend. With Stell, Bob worked on polymerization kinetics, analyzing theories of Flory and Stockmayer and probing the gelation transition, in some ways reminiscent of the Bose–Einstein transition. This led to a long period of working on aggregation and fragmentation with Matthew Ernst and students in Leiden, and his own students Ed McGrady and Richard Dennis Vigil when he became a professor of Chemical Engineering at the University of Michigan in the 1980s. In fragmentation, McGrady and Ziff discovered a 'shattering process' when the rate of fragmentation increases sufficiently when the size decreases, leading to a loss of mass—a kind of opposite problem from gelation [5]. At Stony Brook he also became interested in percolation, and worked with Stell and Peter Cummings on an analysis of hulls of percolation [6], something that anticipated the later work on hulls in the SLE study of percolation. Combining the hull walk with gradient percolation, this led to a paper with Bernard Sapoval on (another) efficient numerical method to determine 2d percolation thresholds [7].

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In those days chemical engineering departments were hiring physicists to modernize their teaching of thermodynamics to include 'statistical thermodynamics' (mechanics). With the advent of powerful computers and simulation software, this has become a major area of research in the engineering field and now the engineering departments produce all the statistical thermodynamics faculty they need.

A few years after Bob arrived at Michigan, a graduate student of fellow professor Erdogan Gulari named Yoav Barshad came to his office with some experimental data on the dynamics of a carbon monoxide-oxygen reaction on a catalyst surface, one of the central reactions in an automotive catalytic converter, and they developed a kinetic Monte Carlo model of it. This became known as the Ziff–Gulari–Barshad (ZGB) model, which, though simple, could explain the multiple steady states and kinetic phase transitions seen in experimental systems. By good fortune, when Bob gave a five-minute talk on it shortly thereafter at a Rutgers Statistical Mechanics conference, Ron Dickman was in the audience, and wrote some nice papers using one- and two-site approximations to solve the behavior; this early work helped to popularize the ZGB model. This work has since been cited over 1000 times. Vladimir Zhdanov, who contributed to this issue, has also modeled this reaction using kinetic Monte Carlo methods.

Bob attended the Rutgers (previously Yeshiva) conferences, run by Joel Lebowitz, off and on since his days as a graduate student in the 1970s. Many important collaborations came from those meetings. When Bob presented his conjecture about the enclosed-area distribution in percolation, John Cardy said 'I think I can calculate that' leading to [8]. John Cardy discusses this in the present volume [9]. It was at the Rutgers meeting that Bob met Peter Kleban, now his good friend, and they started a long collaboration on percolation, again starting with Kleban saying 'I think I can calculate that' to Ziff's observations about excess numbers in percolation. Some of that work was done with Peter's student, Jacob Simmons, who went on to do a post-doc with John Cardy. A mathematics student at Michigan (Steven Flores) also worked with Kleban and Bob, along with Charlie Doering, and Flores wrote many nice papers on percolation and conformal theory, such as crossing problems in different geometries.

In 2005, Bob received an email from a then-graduate student at the University of Chicago, Chris Scullard, who proposed new exact percolation thresholds for a 'martini' lattice (it looks like martini glasses lined up) and related lattices. One of the thresholds was simply $1/\sqrt{2}$, which seemed too simple, but Bob ran a simulation to test it on his laptop while at an Aggregation/Fragmentation conference in Edinburgh, and it did indeed appear correct. Scullard had a proof which was limited to certain lattices, and Bob developed another approach based on the connectivity of triangles (hyperbonds) which could be generalized to more lattices and recover all known exact results—except one, the bow-tie lattice solution discovered by John Wierman. But then that could be included too by generalizing the triangular hypergraph lattice to hypergraphs of triangles (3-bonds) that have a self-dual property [10]. This led to many collaborations with Scullard and Wierman. One remaining question was the exact solution for inhomogeneous checkerboard percolation on the square lattice, whose solution was found by Wu (and verified numerically by Bob); in this case the proof requires use of the isoradial construction of Grimmett and Manolescu, which also follows from duality [11]. While once it was said that exact results in percolation were limited to the few that were studied by Sykes and Essam [12] (bond percolation on the triangular, honeycomb and square lattices, and site percolation on the triangular and kagome lattices), in fact exact results can be found on an infinite number of possible lattices—though some important lattices such as bond percolation on kagome and site percolation on the square lattice have resisted exact solution, and perhaps always will. Note that the Ziff criticality condition 'all connect equals none connect' for a regular triangular hypergraph array was discussed by Chayes and Lei at about the same time [13], a preprint appearing on the arXiv just as Bob was writing up his work.

In a series of papers, Jesper Jacobsen and Chris Scullard developed an algebraic method for locating critical thresholds in two dimensional percolation, finding some thresholds to over 12 digits of accuracy, well beyond the capability of Monte Carlo to test [14–16]. They used a condition on square periodic pieces of the lattice with an effective 'all equals none' condition, or more precisely that a cross-wrapping configuration occurs with the same probability as a no-wrap configuration. The latter is just the probability of a cross-configuration on the dual or matching lattice, and the condition can be seen as saying the cross-wrapping behavior of the two lattices is the same, which is true when the size of the systems is infinite, by universality. In work with Stephan Mertens [17], Bob showed that this generalized 'all equals none' condition for lattices can be cast in terms of clusters on the lattice and matching (or dual) lattice, as a generalization of the work of Sykes and Essam [12].

Another collaboration started with an email was that with Mark Newman, then at the Santa Fe Institute, who had an idea about simulating percolation using the 'union-find' algorithm from computer science. This led to the Newman–Ziff algorithm, which has become one of the most common numerical methods used to study percolation [18, 19]. Shortly thereafter Newman moved to the University of Michigan, and is part of the Center for the Study of Complex Systems, where Ziff is also a member.

To keep track of all the work done on percolation thresholds, Bob started a Wikipedia page on it, with the hope that others would add to it and carry it on. It has grown tremendously (although mainly by Bob's work) and is viewed by about 70 people per day. He also started a page on percolation critical exponents, and continues to think about new methods to find percolation thresholds exactly and approximately.

Bob has enjoyed traveling to various parts of the world, working regularly in China with Youjin Deng at USTC and Deng's ex-student Hao Hu at nearby Anhui University [20], in India with S S Manna [21], and visiting Orsay, France twice (once as a graduate student, and then again about 40 years later working with S Majumdar [22]), Leiden and Utrecht on several occasions in the 1980s working with Matthieu Ernst (an ex-student of E G D Cohen) and others. He visited Chulalongkorn University in Bangkok regularly to teach fluid dynamics in a program connected with the University of Michigan, and worked with Boonyarach Kitiyanan on a paper related to catalysis modeling. In London he worked with Ginestra Bianconi. He has worked with Michał Cieśla from Krakow [23], and recently with Paulo Martins from Brazil and Ron Dickman, on random sequential adsorption problems. With Tania Tomé from Brazil he worked on the SIR model and relations to percolation. He is currently in an active remote collaboration with Greg Huber (Chan-Zuckerberg), Walter Trump (Nuremberg), and Craig Knecht (Nashville) on tilings of sphinx-shaped hexiamonds. He has also enjoyed attending most of the STATPHYS meetings in the past several decades.

Bob has two children David and Anna, and a step-daughter Vera, who now has three children of her own. Anna is in graduate school studying economics, Vera is an accomplished environmental lawyer, and David is a special person who loves going to jazz and other music events regularly with his father. Bob also enjoys long-distance bicycling, bootcamp-style (CrossFit) gym workouts, Latin dancing, and cooking paella. For the latter, he regularly makes trips to Spain to carry out further research.

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