by Marek Biskup and Roberto Schonmann
Thomas M. Liggett (March 29, 1944–May 12, 2020) was a world-renowned probabilist and an influential member of the UCLA Department of Mathematics. He came to UCLA in 1969 as a fresh PhD from Stanford, became a full professor in 1976 and spent altogether 42 years as a regular faculty and nearly 9 years as an emeritus at UCLA. Over this time he achieved remarkable research accomplishments, served in several university functions including being the department chair during 1991-1994, advised 9 PhD students, served as editor-in-chief of a top probability journal and managed probability life at UCLA and in Southern California. He was a speaker at the International Congress of Mathematicians in 1986 and received a number of awards; most notably, a Guggenheim Fellowship in 1997 and memberships in the National Academy of Sciences in 2008 and the American Academy of Arts and Sciences in 2012.
Liggett’s interest in probability was spawned by interactions with Samuel Goldberg during his undergraduate time at Oberlin College. He continued to a PhD program at Stanford where he was further influenced by lectures of probabilist Kai Lai Chung. He ultimately wrote his PhD under the supervision of Samuel Karlin whom he found a better match personally, but he did not find probability research exciting and was even readying himself for a career of a liberal-arts college lecturer, rather than a research mathematician. His advisor urged him to at least call UCLA Math and ask for job application forms. To his surprise, the letter he received in return contained a job offer and this is how he ended up moving to Southern California in 1969.
Perhaps because of the temporary decline of his interest in probability, Liggett’s initial work at UCLA was in functional analysis. In 1971, jointly with Michael Crandall he published what is now known as the Crandall–Liggett Theorem that gives the construction of a semigroup, and thus the solution to a Cauchy problem, for nonlinear generators with a bounded resolvent. Liggett once mentioned that he would have continued working in functional analysis had Chuck Stone, a leading UCLA probabilist at the time, not given him a paper by Frank Spitzer that attempted to set up a theory of Markov processes with interactions. This moment started a flurry of activity that, over the next few years, laid the foundations of a theory of interacting particle systems.
An interacting particle system is a continuous-time Markov process on a countable product of finite sets. The state space is totally disconnected which, at that time, posed technical difficulties that called for a new theory. Unlike earlier attempts that were model-specific, Liggett took a general approach rooted, no surprise, in functional analysis. In a sequence of some 10+ papers, mostly solo or joint with Richard Holley, he found the conditions for the dynamics to be well defined and developed new methods to establish ergodicity. Over his entire career, he continued to study important examples of interacting particle systems; namely, the voter model, the contact process and the exclusion process, and proved a number of deep results about them. The exclusion process is at the core of the recent developments in KPZ universality.
Liggett’s earlier findings on interacting particle systems, along with several other core results in the area are summarized in his 1985 book that influenced generations of probabilists and scientists. But Liggett’s contribution can be found in many other parts of probability as well. In 1983, jointly with Richard Durrett, he characterized fixed points of so-called smoothing transformations. This has recently resurfaced in the studies of multiplicative cascades and chaos measures. In 1985 he found a different formulation of Kingman’s Subadditive Ergodic Theorem that is easier to validate in specific situations. In 1997 he proved, jointly with Roberto Schonmann and Alan Stacey, a stochastic-domination result of finite-range dependent processes by Bernoulli measures which is now an indispensable tool in the studies of intricate percolation problems.
Towards the end of the 1990s, through a connection to the exclusion process, Liggett became interested in negative dependence. This is the property where the occurrence of some natural events makes other natural events less likely to occur. Unlike positive dependence where a theory existed since the 1970s, negative dependence lacked a systematic treatment. A major step towards this goal is Liggett’s joint work with Julius Borcea and Petter Brändén in 2009, where a rich class of measures (called strongly Rayleigh) with negative dependence was introduced and studied. Another remarkable result on the exclusion process is Liggett’s proof in 2010, jointly with Pietro Caputo and Thomas Richthammer, of the so-called Aldous conjecture on the size of the spectral gap of the generators of exclusion processes on graphs.
While working primarily in probability, deep in his heart Liggett was an analyst with formidable problem-solving skills. He would not let a problem go, sometimes thinking about it on and off for years, until he cracked it. He would even experiment with Mathematica at times to find a structure hidden in complicated formulas. While he generally focused on problems of intrinsic value and importance, he would be happy to work on a question just because it posed a challenge to his skills.
Liggett continued working after he retired from his regular position in 2011. He did find teaching too tiring in his last years as a regular faculty but kept pushing on research for several years into his retirement. In late February 2019, just before his 75th birthday conference at IPAM, he came down with pneumonia that left him hospitalized for over a week and then in home care for months. He kept his spirits up for a while, just as he always had when life had presented him with difficulties, but was losing physical strength gradually. The fatigue came with a heavy toll: he could not think creatively any more and that kept him, a thinker for his entire life, extremely frustrated.
Tom Liggett spent his last months in home care with his wife Chris on his side looking after him and keeping him happy as she had over the long years of their marriage. He died peacefully in the evening hours of May 12, 2020, survived by his wife and his children Tim and Amy and their families. He will be dearly missed by everybody who knew him, for he was a great man, mathematician, colleague and friend.
Marek Biskup and Roberto Schonmann are members of the Department of Mathematics at the University of California in Los Angeles.
