# Celebratio Mathematica

## Thomas Milton Liggett

### Tom Liggett and the basic coupling

#### by Pablo A. Ferrari

My first con­tact with Lig­gett’s ideas was in 1981, when his stu­dent En­rique An­djel came to São Paulo, handed me a copy of the pa­per “Coup­ling the simple ex­clu­sion pro­cess” (Ann. Prob­ab­il­ity 4:3 (1976), 339–356) and said: “You should read from here to here”, a couple of pages con­tain­ing this para­graph:

An­oth­er im­port­ant prop­erty of the coupled pro­cess can be seen most clearly by think­ing of it as a pro­cess in which the particles which move on $$S$$ are of four types: $$(0,0)$$, $$(1,1)$$, $$(0,1)$$, $$(1,0)$$. Then some of the trans­itions can be though of as res­ult­ing in the de­struc­tion of the particles of the last two types and their re­place­ment by particles by the first two types. There­fore the “num­ber” (which is usu­ally in­fin­ite) of particles of the first two types can only in­crease.

This early con­tact with the ba­sic coup­ling was in­cred­ibly in­flu­en­tial for me. I used it in my doc­tor­al thes­is and then worked on the simple ex­clu­sion pro­cess for many, many years build­ing on this beau­ti­ful coup­ling con­struc­tion. The $$(1, 0)$$ particles were later called second class particles and played a cru­cial role in the in­ter­play between the simple ex­clu­sion pro­cess, the Bur­gers equa­tion and the last pas­sage per­col­a­tion growth mod­el, in­clud­ing the KPZ boom. Tom’s 1985 in­ter­act­ing particle sys­tems book, called the yel­low book by my young daugh­ters, has been around me ever since.

I first met him per­son­ally at Dart­mouth Col­lege in 1984, dur­ing a work­shop or­gan­ized by Rick Dur­rett. I was a postdoc fel­low at Rut­gers and was look­ing at all those prob­ab­il­ity mon­sters with ad­mir­a­tion. Tom ap­proached and said with a per­fect Ar­gen­tini­an ac­cent “¿Hola, tra­ba­jas con En­rique en São Paulo?” He learned the lan­guage dur­ing his child­hood in Buenos Aires while his fath­er was a pas­tor of the Dis­ciples of Christ church at Za­pi­ola Street, a few blocks away from my cur­rent home. Tom came to the city again in the 90s and was very touched by a vis­it to one of his closest friends from that time.

I have two pa­pers with Tom: one with Ant­o­nio Galves about the time needed by the sym­met­ric simple ex­clu­sion pro­cess to empty a box, and the oth­er with Norio Konno and Vladi­mir Be­l­it­sky with a cor­rel­a­tion in­equal­ity for the con­tact pro­cess. Most of my coau­thors sit with me in front of a black­board for hours wait­ing for an idea to pop up. With Tom it was dif­fer­ent. He would listen to your prob­lem, say that he will think about it and a few days later re­appear with a com­pletely writ­ten solu­tion. He liked solo think­ing, but he was al­ways curi­ous and sup­port­ive of al­tern­at­ive ap­proaches. We were at Les Houches in 1992, par­ti­cip­at­ing in a math­em­at­ic­al phys­ics meet­ing where many talks were very far from prob­ab­il­ity; however Tom at­ten­ded all the talks.

In 2000 Tom gave a talk at the an­nu­al prob­ab­il­ity school EBP (Escola Brasileira de Prob­ab­il­id­ade) in Mam­bu­caba (An­gra dos Re­is, Brasil), and at lunch time some par­ti­cipants were dis­cuss­ing on­line pub­lic­a­tions. A ma­jor cata­strophe could turn off all com­puters, eras­ing elec­tron­ic journ­als and books, but pa­per journ­als would al­ways be ac­cess­ible, ar­gued Tom. Re­mem­ber­ing his com­ment today while look­ing at the yel­low book, I real­ize that he was writ­ing his pa­pers not only for our con­tem­por­ary com­munity, but with an eye to the math­em­at­ic­al read­ers of the fu­ture.

Pablo A. Fer­rari is a Pro­fess­or in the De­part­ment of Math­em­at­ics at the Uni­ver­sid­ad de Buenos Aires in Buenos Aires, Ar­gen­tina.