Celebratio Mathematica

Thomas Milton Liggett

On interactions with Professor Tom Liggett

by Sunder Sethuraman

Al­though I was not a form­al stu­dent of Pro­fess­or Tom Lig­gett, through vari­ous points of my ca­reer Tom was an im­port­ant ment­or. Al­ways free to share know­ledge, Tom was one to take time to an­swer ques­tions and give care­ful ad­vice when asked. Many ca­reers, in­clud­ing mine, de­rive in part from his found­a­tion­al re­search in the field of in­ter­act­ing particle sys­tems.

My first in­tro­duc­tion to Tom was through his book In­ter­act­ing Particle Sys­tems (1985). I was asked to go through it in 1992 as a gradu­ate stu­dent at NYU start­ing PhD thes­is re­search on ex­clu­sion and zero-range pro­cesses with Raghu Varadhan. Tom’s book was a con­stant com­pan­ion then, of­ten turned to Chapter 8, “The Ex­clu­sion Pro­cess”.

My thes­is prob­lem was to un­der­stand for which func­tions \( f \) the lim­it­ing vari­ance of the scaled ad­dit­ive func­tion­al, \( \smash{\frac{1}{\sqrt{t}}}\int_0^t f(\eta_s)\,ds \), with re­spect to these in­ter­act­ing particle sys­tems was fi­nite. Such a res­ult relates to the scal­ing be­ha­vi­or of a tagged, or dis­tin­guished, particle in these mod­els, a con­cern of in­terest to Tom as well, who wrote on as­pects of tagged particles in In­ter­act­ing Particle Sys­tems and also in his book Stochast­ic In­ter­act­ing Sys­tems: Con­tact, Voter and Ex­clu­sion Pro­cesses (1999). I first cor­res­pon­ded with Tom in 1994, in the last year of my PhD, ask­ing if he would serve as the primary ad­visor for an NSF postdoc ap­plic­a­tion. Tom was kind to agree, and a pro­pos­al was writ­ten, but at the end the po­s­i­tion didn’t ma­ter­i­al­ize. We kept in touch though and later, while at Iowa State Uni­versity, it was an hon­or to in­vite Tom to speak at the “Ames Work­shop on Particle Sys­tems”, April 2001, or­gan­ized to­geth­er with Timo Seppäläin­en.

Tom’s en­ergy, drive, and op­tim­ism were an in­spir­a­tion. He once en­cour­aged me early on when I was stuck in a re­search prob­lem to in­vest­ig­ate wheth­er a term or small ex­ample could be com­puted, point­ing out that such an ef­fort might lead to a bet­ter out­look.

In a more re­cent en­counter, Tom was ex­cited to dis­cuss his work on neg­at­ive as­so­ci­ation and the ex­clu­sion pro­cess and his 2009 pa­pers “Dis­tri­bu­tion­al lim­its for the sym­met­ric ex­clu­sion pro­cess” and “Neg­at­ive de­pend­ence and the geo­metry of poly­no­mi­als” with Bor­cea and Brändén. He felt there might be ap­plic­a­tion in what I was do­ing at the time. In­deed, these in­ter­est­ing res­ults helped with some es­tim­a­tion in my pa­per, “Large de­vi­ations for the cur­rent and tagged particle in 1D nearest-neigh­bor sym­met­ric simple ex­clu­sion” with Varadhan.

Like many, I was ex­cited to at­tend Tom’s 75th birth­day con­fer­ence two years ago at UCLA in 2018, and was look­ing for­ward to chat­ting more with him again. Al­though he couldn’t be there in per­son, the oc­ca­sion was a cel­eb­ra­tion of his sci­entif­ic and per­son­al im­pact, with a range of par­ti­cipants, from stu­dents to seni­or pro­fess­ors. At the ban­quet, it was won­der­ful to talk with Chris Lig­gett and to meet Tim and Amy, those to whom In­ter­act­ing Particle Sys­tems was ded­ic­ated.

Sun­der Se­th­ura­man is a pro­fess­or in the De­part­ment of Math­em­at­ics at the Uni­versity of Ari­zona.