Celebratio Mathematica

Thomas Milton Liggett

In memoriam: Thomas M. Liggett

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.