Harry Kesten (1931–2019): APersonal and Scientific TributeGeoffrey R. Grimmett and Gregory F. Lawler

Figure 1. Harry Kesten in White Hall, Cornell University,around 1980.

The mathematical achievements of Harry Kesten since themid-1950s have revolutionized probability theory as asubject in its own right and in its associations with as-pects of algebra, analysis, geometry, and statistical physics.Through his personality and scientific ability, he hasframed the modern subject to a degree exceeded by noother.

Harry inspired high standards of honesty, modesty, andinformality, and he played a central part in the creation ofa lively and open community of researchers.

Geoffrey R. Grimmett is professor emeritus of mathematical statistics at Cam-bridge University, and professor of applied probability at the University of Mel-bourne. His email address is grg@statslab.cam.ac.uk.Gregory F. Lawler is the George Wells Beadle Distinguished Service Professorat the University of Chicago. His email address is lawler@math.uchicago.edu.

Communicated by Notices Associate Editor Daniela De Silva.

For permission to reprint this article, please contact:reprint-permission@ams.org.

DOI: https://doi.org/10.1090/noti2100

Biographical NotesEarly life. Harry Kesten’s early life was far from tran-quil. His already migrant father migrated once again fromGermany following Hitler’s appointment as chancellor in1933. Harry survived the war under the protection of anon-Jewish family in the Netherlands. His parents diednaturally during and immediately following the war. In1952, he recognized his likely future as a mathematician.

Harry was born in Duisburg, Germany, on 19 No-vember 1931 to Michael and Elise Kesten. “Hamborn(Hochfeld)” is listed as his place of birth in both a Germanand a Polish document. Harry had Polish citizenship frombirth via his father, and he retained this until his Americannaturalization in 1962.

He was the only child of Michael Kesten (born inPodołowoczyska, [5], then in Galicia and now Ukraine, 22November 1890) and Elise Abrahamovich (born in Char-lottenburg, Berlin, 4 October 1905). The Kesten familyname featured prominently in the affairs of the substan-tial Jewish population of Podołowoczyska in the late nine-teenth century. Michael moved from Galicia to Berlin,where he and Elise were married in Charlottenburg on 23May 1928. Amos Elon has written eloquently in [4] ofthe Jewish community in Berlin up to 1933, the year inwhich the Kesten family moved to Amsterdam, perhaps asa member of a group of Jewish families.

German military forces attacked and invaded theNetherlands in May 1940. Shortly afterwards, Harry wasoffered protection by a non-Jewish Dutch family resid-ing in Driebergen near Utrecht, with whom he lived un-til the Dutch liberation in May 1945. His father was hid-den in the same village, and they could be in occasionalcontact throughout the occupation. During that periodHarry attended school in a normal way ([3]). Elise died ofleukemia in Amsterdam on 11 February 1941, andMichaeldied in October 1945, probably in Groningen of cancer.

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Later in life, Harry kept in touch with his Dutch family,and would visit them whenever he was in the Netherlands.

Harry moved back to Amsterdam at the end of the warto live with an older married cousin who had been borna Kesten, and who had survived the war in Switzerland. Itwas during the period between 1945 and graduation fromhigh school in 1949 that his attachment to Orthodox Ju-daism developed, and this was to remain with him for therest of his life.

The earliest surviving indication of Harry’s scientificability is found in his school report on graduation from theGeneral Secondary High School in 1949. His six grades inthe six given scientific topics (including mathematics) arerecorded as five 10s and one 9. In languages, they were7 (Dutch), 7 (French), 8 (English), 10 (German). Therewere three subjects in which his mark was a mere 6 (“sat-isfactory”), including handwriting and physical education.In later life, Harry was very active physically, and was keento run, swim, hike, and to ski cross-country, usually withfriends and colleagues.

Figure 2. Harry Kesten, cross-country skiing with Rob andMargriet van den Berg, Ithaca, 1991.

Following his uncle’s advice to become a chemical engi-neer, Harry entered the University of Amsterdam in 1949to study chemistry ([3]). This was not altogether success-ful, and Harry took a particular dislike to laboratory work.He moved briefly to theoretical physics before settling onmathematics. From 1952 to 1956 he had a half-time assist-antship in the statistical department of the MathematicalCentre (now the CWI), Amsterdam, under the supervisionof David van Dantzig (known for his theory of collectivemarks) and Jan Hemelrijk. He shared an office with fellowstudent Theo (J. Th.) Runnenburg, with whom he wrotehis first papers on topics in renewal theory and queueingtheory. The pair of papers [20] are notable, since they are

probably related to the master’s (almost, in a sense, doc-toral) thesis that Harry wrote in 1956.

It was around this time that he met his wife-to-be, Do-raline Wabeke, who worked in the Mathematical CentreLibrary while studying interior design at evening school.Middle years. Mark Kac visited Holland in 1955, andHarry had the opportunity to meet him at the Mathemati-cal Centre. He wrote to Kac in January 1956 to enquire ofa graduate fellowship at Cornell University to study prob-ability theory, perhaps for one year. Van Dantzig wrote toKac in support, “. . . I have not the slightest doubt that, ifyou grant him a fellowship, you will consider the moneywell spent afterwards.”

A Junior Graduate Fellowship was duly arranged with astipend of $1,400 plus fees, and Harry joined the mathe-matics graduate program at Cornell that summer, travelingon a passport issued by the International Refugee Organi-zation. His fellowship was extended to the next academicyear 1957–58 with support from Mark Kac: “Mr. Kestenis. . . the best student we have had here in the last twentyyears.. . . one of these days we will indeed be proud of hav-ing helped to educate an outstanding mathematician.”

He defended his PhD thesis at the end of that year, onthe (then) highly novel topic of random walks on groups.This area of Harry’s creation remains an active and fruitfularea of research at the time of this memoir.

Doraline followed Harry to the USA in 1957 under theauspices of “The Experiment of International Living,” andtook a position in Oswego, NY, about seventy-five milesnorth of Ithaca on Lake Ontario. As a result of the At-lantic crossing she became averse to long boat journeys,and never traveled thus again. She studied and convertedto Judaism with a rabbi in Syracuse, and the couple wasmarried in 1958.

Harry and Doraline moved to Princeton in 1958, whereHarry held a (one-year) instructorship in the company ofHillel Furstenberg. For the following academic year, he ac-cepted a position at the Hebrew University of Jerusalem.Harry was interested in settling in Israel, and wanted to tryit out, but there were competing pressures from Cornell,who wished to attract him back to Ithaca, and from Dora-line’s concerns about practical matters. They postponed adecision on the offer of an assistant professorship at Cor-nell (with a standard nine hours/week teaching load), opt-ing instead to return for the year 1961–62 on a one-yearbasis.

It was during that year that Harry and Doraline decidedto make Ithaca their home. Harry was promoted to therank of associate professor in 1962, and in 1965Harry andDoraline celebrated both his promotion to full professorand the birth of their only child, Michael. Harry stayed at

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Cornell for his entire career, becoming emeritus professorin 2002.

Harry, Doraline, and Michael lived for many years withtheir cats at 35 Turkey Hill Road, where visitors would bewelcomed for parties and walks.

Unsurprisingly, many invitations arrived, and he wouldinvariably try to oblige. Harry traveled widely, and paidmany extended visits to universities and research centers inthe USA and abroad, frequently accompanied by Doraline,and in earlier years by Michael.

For some years the principal events in Cornell proba-bility included the biweekly five-mile runs of Harry withFrank Spitzer, and able-bodied visitors were always wel-come. WhenHarry’s knees showed their age, he spent timeswimming lengths in Cornell’s Teagle pool, usually in thenow passe men-only sessions. He worked on his problemswhile swimming.

Figure 3. Harry Kesten with Geoffrey Grimmett in Kendal atIthaca, 2011.

Later years. Harry maintained his research activities andcollaborations beyond his retirement from active duty in2002. He spoke at the Beijing ICM that year on the sub-ject of percolation, finishing with a slide listing individu-als who had been imprisoned in China for the crime ofexpressing dissent ([18]).

He was awarded an honorary doctorate at the Univer-site de Paris-Sud in 2007, shortly following his diagnosis ofParkinson’s disease. Harry and Doraline sold their houseand moved in 2008 into the Kendal retirement home, a

“home away fromhome” for numerous retired Cornell aca-demics. Doraline developed Alzheimer’s disease and diedat Kendal on 2 March 2016, followed by Harry from com-plications of Parkinson’s disease on 29 March 2019. A vol-ume on percolation remained on his bedside table untilthe end.Awards of distinction. From among the awards made toHarry Kesten, mention is made here of the Alfred P. SloanFellowship (1963), the Guggenheim Fellowship (1972),the Brouwer Medal (1981), the SIAM George Polya Prize(1994), and the AMS Leroy P. Steele Prize for LifetimeAchievement (2001). Harry delivered the Wald Memo-rial Lectures of the IMS (1986), and was elected a Corre-spondent of the Royal Netherlands Academy of Arts andSciences (1980), a Member of the National Academy ofSciences (1983), and of the American Academy of Artsand Sciences (1999). He was elected an Overseas Fel-low of Churchill College, Cambridge (1993), and wasawarded an Honorary Doctorate of the Universite de Paris-Sud (2007).

He was an invited lecturer at three ICMs—Nice (1970),Warsaw (1983), and Beijing (2002)—and he spoke at theHyderabad ICM (2012) on the work of Fields MedalistStanislav Smirnov. Hewas amember of the inaugural classof Fellows of the AMS in 2013.

Personality and InfluenceProbability theory gained great momentum in the secondhalf of the twentieth century. Exciting and beautiful prob-lems were formulated and solved, and connections withother fields of mathematics and science, both physical andsocio-economic, were established. The general area at-tracted a large number of distinguished scientists, and itgrew in maturity and visibility. Harry was at the epicenterof the mathematical aspects of this development. He con-tributed new and often startling results at the leading edgeof almost every branch of probability theory.

Despite an occasionally serious aspect, he was a verysociable person who enjoyed his many scientific collabo-rations and was a popular correspondent. His archive ofpapers (now held by Cornell University) reveals a wealthof letters exchanged with many individuals worldwide,and every serious letter received a serious reply, frequentlyproposing solutions to the problems posed. He was espe-cially keen to discuss and collaborate with younger people,and he played a key role through his achievements and per-sonality in bringing them into the field.

Harry commanded enormous respect and affectionamongst those who knew him well. He displayed an un-compromising honesty, tempered by humanity, in bothpersonal and professional matters. This was never clearerthan in his opposition to oppression, and in his public

824 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6

Figure 4. Harry Kesten and Frank Spitzer in 1970.

support for individuals deprived of their positions, or evenliberty, for expressing their beliefs or needs.

Harry loved hard problems. Supported by an extraordi-nary technical ability and a total lack of fear, he gained ajust reputation as a fearsome problem solver. His workoften exceeded the greatest current expectations, and itcould be years before the community caught up with him.When the going became too tough for the rest of us, hewould simply refuse to give in. The outcome is 196 workslisted on MathSciNet, almost every one of which containssome new idea of substance. An excellent sense of Harryas a mathematician may be gained by reading the first twopages of Rick Durrett’s appreciation [2], published in 1999to mark 40+ years of Harry’s mathematics.

Scientific WorkMost areas of probability theory have been steered, evenmoulded, by Harry, and it is not uncommon to attend con-ferences in which a majority of speakers refer to his workas fundamental to their particular topics. In this memoir,we do not aim at a comprehensive survey but, instead, toselect and describe some high points. Our selection is per-sonal by necessity, and readers desirous of a more compre-hensive account are referred to [2,7].Random walks on groups. Harry’s PhD work was on thetopic of random walks on groups. So-called “simple ran-dom walk” takes random jumps of size ±1 about the lineℤ. The domain ℤmay be replaced by either ℤ𝑑 or ℝ𝑑, andthe unit moves replaced by a general family of indepen-dent and identically distributed displacements. Harry’sspace was algebraic rather than Euclidean. He considereda countable group 𝐺, a symmetric (𝑝(𝑥) = 𝑝(𝑥−1)) prob-ability distribution on a generating set of 𝐺, and definedrandom walk on 𝐺 as the process that moves at each stepfrom 𝑦 to 𝑦𝑥 with probability 𝑝(𝑥). Let 𝑞2𝑛 denote the

probability that the random walk returns to its startingpoint after 2𝑛 steps. For example, the usual random walkon ℤ𝑑 has 𝑞2𝑛 ∼ 𝑐𝑑𝑛−𝑑/2, while for the free group on two

generators we have 𝑞2𝑛 ≈ ( 34)𝑛.

Harry showed for a general countably infinite group 𝐺that the quantity

Λ(𝐺, 𝑝) ∶= lim𝑛→∞

𝑞1/2𝑛2𝑛

equals both the spectral radius and the maximal value ofthe spectrum of the associated operator on 𝐿2(𝐺) given bythe random walk. He proved that the equality Λ(𝐺, 𝑝) = 1is a property of the group𝐺 and not of the particulars of thetransition probabilities 𝑝, and if it holds wewriteΛ(𝐺) = 1.The so-called “Kesten criterion for amenability” states thatΛ(𝐺) = 1 if and only if 𝐺 is amenable.1 This remarkablecharacterization of amenability may be viewed as a fairlyearly contribution to the currently important area of geo-metric group theory (see [1]).Products of random matrices. One of the earliest papersin the now important field of random matrices is [6] byFurstenberg and Kesten, written by two Princeton instruc-tors in 1958–59. They were motivated by the 1954 workof Bellman, studying the asymptotic behavior of the prod-uct of 𝑛 independent, random 2 × 2matrices, and they de-rived substantial extensions of Bellman’s results. This nowclassical paper [6] has been very influential and is muchcited, despite having proved unwelcome at the authors’first choice of journal. It deals with products of randommatrices, in contrast to most of the modern theory whichis directed towards spectral properties.

Consider a stationary sequence 𝑋1, 𝑋2, … of random 𝑘×𝑘 matrices, and let

𝑌𝑛 = (𝑦𝑛𝑖𝑗) = 𝑋𝑛𝑋𝑛−1⋯𝑋1.In an analysis termed by Bellman “difficult and ingenious,”Furstenberg and Kesten proved a law of large numbers anda central limit theorem. Firstly, if 𝑋 is ergodic, the limit𝐸 ∶= lim𝑛→∞ 𝑛−1 log ‖𝑌𝑛‖1 exists a.s. Secondly, subjectto certain conditions, the limit of 𝑛−1𝔼(log 𝑦𝑛𝑖𝑗) exists, and𝑛−1/2(log 𝑦𝑛𝑖𝑗 − 𝔼(log 𝑦𝑛𝑖𝑗)) is asymptotically normally dis-tributed.

Although they used subadditivity in the proofs, they didnot anticipate the forthcoming theory of subadditive sto-chastic processes, initiated in 1965 by Hammersley andWelsh to study first-passage percolation, which would oneday provide a neat proof of some of their results.

Harry returned in 1973 to a study of products of ran-dom matrices arising in stochastic recurrence relations. In

1Of the various equivalent definitions of amenability, the reader is remindedthat a discrete group 𝐺 is amenable if there exists a sequence 𝐹𝑛 of finite subsetssuch that, for 𝑔 ∈ 𝐺, |(𝑔𝐹𝑛)△𝐹𝑛|/|𝐹𝑛| → 0 as 𝑛 → ∞.

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the one-dimensional case, it was important to understandthe tail behavior of a random variable 𝑌 satisfying a sto-

chastic equation of the form 𝑌 𝑑= 𝑀𝑌 + 𝑄; that is, for ran-dom variables 𝑀 and 𝑄 with given distributions, 𝑌 and𝑀𝑌+𝑄 have the same distribution. He proved in [11] thatsuch 𝑌 are generally heavy-tailed in that ℙ(𝑌 > 𝑦) decaysas a power law as 𝑦 → ∞. This work has generated a veryconsiderable amount of interest since in probability, sta-tistics, and mathematical finance.Random walks and Levy processes. Harry’s early years atCornell marked a heyday for the theory of random walk.His colleague Frank Spitzer hadwritten his Springermono-graph Principles of Random Walk, and, together, they andtheir collaborators did much to further the field. The clas-sical definition of random walk is as a sum 𝑆𝑛 = 𝑥 + 𝑋1 +⋯+ 𝑋𝑛 where 𝑥 is an initial position and 𝑋1, 𝑋2, … are in-dependent and identically distributed. When the distribu-tions are nice, the theory parallels that of the continuouspotential theory of the Laplacian. When the hypotheses ondistributions are weakened, some but not all such proper-ties persist.

In an early sequence of papers, Harry proved a familyof ratio limit theorems for probabilities associated withrandom walks on ℤ𝑑. Here are two examples, taken fromjoint work with Spitzer and Ornstein. Let 𝑇 denote thefirst return time of 𝑋 to its starting point. Then ℙ0(𝑇 >𝑛 + 1)/ℙ0(𝑇 > 𝑛) → 1 as 𝑛 → ∞, and this may be used toshow that the limit

𝑎(𝑥) = lim𝑛→∞

ℙ𝑥(𝑇 > 𝑛)ℙ0(𝑇 > 𝑛)

exists and equals the potential kernel (fundamental solu-tion of the corresponding discrete Laplacian) at 𝑥, that is,

𝑎(𝑥) =∞∑𝑛=0

[ℙ0(𝑋𝑛 = 0) − ℙ𝑥(𝑋𝑛 = 0)], 𝑥 ≠ 0.

This result requires no further assumptions on the randomwalk. (The subscript 𝑥 on ℙ𝑥 denotes the starting point.)

A Levy process is a random process in continuous timewith stationary independent increments. Let 𝑋 be a Levyprocess on ℝ𝑑. The fundamental question arose throughwork of Neveu, Chung, Meyer, and McKean of decidingwhen the hitting probability ℎ(𝑟) by 𝑋 of a point 𝑟 ∈ ℝ𝑑

satisfies ℎ(𝑟) > 0. Harry solved this problem in his extra-ordinary AMS Memoir of 1969, [10]. The situation is sim-plest when 𝑑 = 1, for which case Harry showed that (apartfrom special cases) ℎ(𝑟) > 0 if and only if the so-calledcharacteristic exponent of 𝑋 satisfies a certain integral con-dition.

The expression “random walk” is sometimes usedloosely in the context of interesting and challengingproblems, arising for instance in physical and biological

models, which lack the full assumptions of independenceand identical distribution of jumps. Harry respondedto the challenge to attack many of these difficult prob-lems for which the current machinery was not sufficient.He combined his mastery of classical techniques with his“problem-solving” ability to develop new ideas for suchnovel topics.

A significant variant of the classical random walk is the“random walk in random environment” (RWRE). In theone-dimensional case, this is given by (i) sampling ran-dom variables for each site that prescribe the transitionprobabilities when one reaches the site, and (ii) perform-ing a random walk (or more precisely a Markov chain)with those transition probabilities. RWRE is a Markovchain given the environment, but it is not itself Markovianbecause, in observing the process, one accrues informationabout the underlying random environment.

One of the first RWRE cases considered was a near-est neighbor, one-dimensional walk, sometimes called abirth-death chain on ℤ. Let 𝛼𝑥 denote the probability thatthe random walk moves one step rightward when at po-sition 𝑥 (so that 𝛽𝑥 = 1 − 𝛼𝑥 is the probability of movingone step leftward). We assume the 𝛼𝑥 are independent andidentically distributed. In the deterministic environmentwith, say, 𝛽𝑥 = 𝛽 < 1

2for all 𝑥, as 𝑛 → ∞, 𝑋𝑛/𝑛 → 1 − 2𝛽;

when 𝛽 = 12, the walk returns to the origin infinitely often.

For the RWRE (with random 𝛼𝑥), the distributional proper-ties of the ratio 𝛽0/𝛼0 are pivotal for determining whetheror not 𝑋𝑛 →∞; a straightforward birth-death argument in-dicates that 𝑋𝑛 →∞ if 𝔼[log(𝛽0/𝛼0)] < 0. The regime with𝔼[log(𝛽0/𝛼0)] < 0 and 𝔼[𝛽0/𝛼0] > 1 turns out to be interest-ing. It was already known for this regime that 𝑋𝑛 →∞ but𝑋𝑛/𝑛 → 0. With Kozlov and Spitzer [19], Harry showed inthis case that the rate of growth of 𝑋𝑡 is essentially deter-mined by the value of the parameter 𝜅 defined by

𝔼 [(𝛽0/𝛼0)𝜅] = 1.

In particular, if 𝜅 < 1, then 𝑋𝑛 has order 𝑛𝜅. It turns outthat the limit distribution of 𝑛−𝜅𝑋𝑛 may be expressed interms of a stable law with index 𝜅. This result answered aquestion of Kolmogorov.

The above is an early contribution to the broad areaof RWRE, and to the related area of homogenization ofdifferential operators with random coefficients. The one-dimensional case is special (roughly speaking, because thewalk cannot avoid the exceptional regions in the environ-ment), and rather precise results are now available, includ-ing Sinai’s proof that 𝑋𝑛/(log 𝑛)2 converges weakly (to adistribution calculated later by Harry). RWRE in higher di-mensions poses a very challenging problem. This processis more diffusive than its one-dimensional cousin, and its

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study requires different techniques. Major progress hasbeen made on it by Bricmont, Kupiainen, and others.Self-avoiding walk. A self-avoiding walk (or SAW) on the 𝑑-dimensional hypercubic lattice ℤ𝑑 is a path that visits novertex more than once. SAWs may be viewed as a simplemodel for long-chain polymers, and the SAW problem isof importance in physics as well as in mathematics.

Let 𝜒𝑛 be the number of 𝑛-step SAWs starting at theorigin. The principal SAW problems are to establish theasymptotics of 𝜒𝑛 as 𝑛 → ∞, and to determine the typi-cal radius of a SAW of length 𝑛. Hammersley and Mortonproved in 1954 that there exists a “connective constant”𝜅 such that 𝜒𝑛 = 𝜅𝑛(1+o(1)) as 𝑛 → ∞. The finer asymp-totics of 𝜒𝑛 have proved elusive, especially in dimensions𝑑 = 2, 3.

Harry’s ten-page paper [9] from 1963 contains a the-orem and a technique; the theorem is essentially unim-proved, and the technique has frequently been key to thework of others since. His main result is the ratio limit the-orem that 𝜒𝑛+2/𝜒𝑛 → 𝜅2 as 𝑛 → ∞. It remains an openproblem to prove that𝜒𝑛+1/𝜒𝑛 → 𝜅. His technique is an ar-gument now referred to as “Kesten’s pattern theorem.” Toparaphrase Frank Spitzer from Mathematical Reviews, theidea is that any configuration of 𝑘 steps which can occurmore than once in an 𝑛-step SAW has to occur at least 𝑎𝑛times, for some 𝑎 > 0, in all but “very few” such SAWs.

The pattern theorem is proved using a type of pathsurgery that has been useful in numerous other contextssince. The proof is centered around an “exponential esti-mate” of a general type that made powerful appearancesin various different settings in Harry’s later work.

One of the most prominent current conjectures in SAWtheory is that SAW in two dimensions converges, in anappropriate limit, to a certain Schramm–Loewner evolu-tion (namely, SLE8/3). If one could show that the limitexists and exhibits conformal invariance, then it wouldbe known that the limit must be SLE8/3. Although thebehavior of this SLE is now understood fairly precisely([22]), and the analogues of the finite asymptotics of 𝜒𝑛and the typical length of a SAW are known for the contin-uous model, the problems of establishing that the discreteSAW has a limit, and showing that the limit is conformallyinvariant, remain wide open.Diffusion limited aggregation. Diffusion limited aggre-gation (DLA) is a growth model introduced by Witten andSander. The model may be defined in general dimensions𝑑, but we concentrate here on the case 𝑑 = 2.

Let𝐴0 be the origin of the square latticeℤ2. Conditionalon the set𝐴𝑛, the set𝐴𝑛+1 is obtained by starting a randomwalk “at infinity” and stopping it when it reaches a pointthat is adjacent to 𝐴𝑛, and then adding that new point to

Figure 5. A simulation of diffusion limited aggregation in twodimensions.

the existing 𝐴𝑛. The concept of random walk from infin-ity can be made precise using harmonic measure, and thehard problem is then to describe the evolution of the ran-dom sets 𝐴𝑛. Computer simulations suggest a random,somewhat tree-like, fractal structure for 𝐴𝑛 as 𝑛 → ∞. In-deed, assigning a “fractal dimension” to such a set is a sub-tle issue, but a start is made by trying to find the exponent𝛼 such that the diameter of 𝐴𝑛 grows like 𝑛𝛼. Since 𝐴𝑛 isa connected set of 𝑛+ 1 points, we have the trivial bounds12≤ 𝛼 ≤ 1.Harry wrote a short paper [15] containing a beautiful

argument showing that 𝛼 ≤ 23. For many readers, this

seemed a good start to the problem, andmuch subsequenteffort has been invested in seeking improved estimates.Unfortunately, no one has yet made a substantial rigorousimprovement to Harry’s bound. There are a number of pla-nar models with diffusive limited growth, and it is openwhether they are in the same universality class. On theother hand, “Kesten’s bound” is one property they have incommon.

Harry’s argument is simple, but in order to completeit, he needed a separate lemma ([17]) about planar ran-dom walks which is a discrete analogue of a theorem fromcomplex variables due to Beurling. Kesten’s lemma hasitself proved an extremely useful tool over the last thirtyyears in the development of the theory of conformally in-variant limits of two-dimensional randomwalks and otherprocesses. Although the original theorem of Beurling was

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phrased in terms of complex analysis, both the continu-ous version of Beurling and the discrete lemma of Kestenbecomemost important in probabilistic approaches to prob-lems. While Kesten figured out what the right answershould be in his discrete version, hewas fortunate to have acolleague, Clifford Earle, who knew Beurling’s result andcould refer Harry to a proof that proved to be adaptable(with work!) to the discrete case.Branching processes. The branching process (sometimescalled the Galton–Watson process) is arguably the mostfundamental stochastic model for population growth. In-dividuals produce a random number of offspring. Theoffspring produce offspring similarly, and so on, with dif-ferent family-sizes being independent and identically dis-tributed. The first question is whether or not the popu-lation survives forever. A basic fact taught in elementarycourses on stochastic processes states that the populationdies out with probability 1 if and only if 𝜇 ≤ 1, where 𝜇is the mean number of offspring per individual. (Thereis a trivial deterministic exception to this, in which eachindividual produces exactly one offspring.)

One of Harry’s best known results is the Kesten–Stigumtheorem [21] for the supercritical case 𝜇 > 1; it was provedin the more general situation with more than one type ofindividual, but we discuss here the situation with only onepopulation type. If 𝑋𝑛 denotes the population size of the𝑛th generation, we have 𝔼[𝑋𝑛] = 𝜇𝑛𝑋0, and with some(computable) probability 𝑞 > 0, the population survivesforever. One might expect that, for large 𝑛, 𝑋𝑛 ∼ 𝐾∞𝜇𝑛with 𝐾∞ a random variable determined by the growth ofthe early generations. In other words, once the popula-tion has become large, we should be able to approximateits growth by the deterministic dynamics 𝑋𝑛+1 ∼ 𝜇𝑋𝑛. Ifthis were true, we would write 𝐾∞ = lim𝑛→∞ 𝐾𝑛 where𝐾𝑛 = 𝑋𝑛/𝜇𝑛. The process 𝐾𝑛 is a martingale, and the mar-tingale convergence theorem implies that 𝐾𝑛 converges al-most surely to some limit 𝐾∞. However, it turns out to bepossible that 𝐾∞ = 0 a.s., even though the process survivesforever with a strictly positive probability.

Let 𝐿 be a random variable with the family-size distri-bution, so that 𝜇 = 𝔼[𝐿]. It was a problem of some impor-tance to identify a necessary and sufficient moment condi-tion for the statement that 𝔼[𝐾∞] = 𝑋0. Kesten and Stigumshowed that this condition is that 𝔼[𝐿 log+ 𝐿] < ∞.

Harry was inevitably attracted by the critical branchingprocess (with 𝜇 = 1). Amongst his numerous resultsare the well-known necessary and sufficient conditions(proved with Ney and Spitzer) for the so-called Yaglomand Kolmogorov laws,

ℙ(𝑋𝑛 > 0) ∼ 𝑐𝑛 , ℙ(𝑋𝑛 > 𝑛𝑥 | 𝑋𝑛 > 0) → 𝑒−𝑐′𝑥.

Figure 6. A simulation of bond percolation on ℤ2 with 𝑝 = 0.51.

We return later to his work on random walk on the criticalfamily tree conditioned on non-extinction.Percolation. The percolation model for a disorderedmedium was pioneered by Hammersley in the 1950s/60s,and it has since become one of the principal objects inprobability theory. In its simplest form, each edge of thesquare lattice is declared open with probability 𝑝 and oth-erwise closed, different edges having independent states.How does the geometry of the open graph vary as 𝑝 in-creases, and in particular for what 𝑝 does there exist aninfinite open cluster?

It turns out that there exists a critical probability 𝑝c suchthat an infinite open cluster exists if and only if 𝑝 > 𝑝c. Theso-called “phase transition” at 𝑝c is emblematic of phasetransitions in mathematical physics. Percolation theory isfrequently used directly in the study of other systems, andit has led to the development of a number of powerful in-sights and techniques.

As Harry wrote in the preface of his 1982 book [13],

Quite apart from the fact that percolation theoryhad its origin in an honest applied problem… , it isa source of fascinating problems of the best kind amathematician can wish for: problems which areeasy to state with a minimum of preparation, butwhose solutions are (apparently) difficult and re-quire new methods. At the same time many of theproblems are of interest to or proposed by statisti-cal physicists and not dreamt upmerely to demon-strate ingenuity.

They certainly require ingenuity to solve, as demonstratedin Harry’s celebrated proof that 𝑝c =

12for the square lat-

tice problem, published in 1980 ([12], see Figure 6). Harry

828 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6

Figure 7. Harry Kesten, Rudolf Peierls, and Roland Dobrushinin New College, Oxford, 1993.

proved that 𝑝c ≤12, thereby complementing Harris’s ear-

lier proof that 𝑝c ≥12. His paper, and the book [13] that

followed, resolved this notorious open problem, and in-vigorated an area that many considered almost impossiblymysterious.

Harry’s book [13] was a fairly formidable work contain-ing many new results for percolation in two dimensions,set in quite a general context. He was never frightened bytechnical difficulty or complication, and entertained simi-lar standards of the reader. This project led in a natural wayto his important and far-sighted work [16] on scaling rela-tions and so-called “arms” at and near the critical point,which was to prove relevant in the highly original studyinitiated by Schramm and developed by Smirnov and oth-ers on conformal invariance in percolation.

Write 𝐶 for the open cluster containing the origin. Ac-cording to scaling theory, macroscopic functions, suchas the percolation probability 𝜃 and mean cluster size 𝜒,given by

𝜃(𝑝) = ℙ𝑝(|𝐶| = ∞), 𝜒(𝑝) = 𝔼𝑝|𝐶|,

have singularities at 𝑝c of the form |𝑝 − 𝑝c| raised to anappropriate power called a “critical exponent.” In similarfashion, when 𝑝 = 𝑝c, several random variables associatedwith the open cluster at the origin have power-law tail be-haviors of the form 𝑛−𝛿 as 𝑛 → ∞, for suitable criticalexponents 𝛿. The set of critical exponents describes the na-ture of the singularity and they are characteristics of themodel. They are not, however, independent variables, inthat they satisfy the “scaling relations” of statistical physics.It is an open problem to prove almost any of the above ingeneral dimensions.

In the special case of two dimensions, the proof ofexistence of critical exponents had to wait beyond the

Figure 8. Site percolation on the triangular lattice 𝕋. The redpath is a black/white interface. Smirnov proved Cardy’sformula, which states in this context that the hitting point ofthe interface on the bottom side is asymptotically uniformlydistributed.

invention of SLE by Schramm around 2000, and the proofof Cardy’s formula by Smirnov in 2001 (illustrated in Fig-ure 8), and is the work of several individuals includingLawler and Werner. In a precursor [16] of that, Harryproved amongst other things that, conditional on the ex-istence of certain exponents, certain others must also existand a variety of scaling relations ensue. In this work, heintroduced a number of techniques that have been at theheart of understanding the problem of conformal invari-ance.

Correlation inequalities are central to the theory of dis-ordered systems in mathematics and physics. The highlynovel BK (van den Berg/Kesten) inequality plays a key rolein systems subjected to a product measure such as perco-lation. Proved in 1985, this inequality is a form of nega-tive association, based around the notion of the “disjointoccurrence” of two events. It is a delicate and tantalizingresult.

When 𝑝 > 𝑝c, there exists a.s. at least one infinite opencluster, but how many? This uniqueness problem was an-swered by Harry in joint work with Aizenman and New-man. He tended to downplay his part in this work, but hisfriends knew him better than to take such protestations atface value. Their paper was soon superseded by the elegantargument of Burton and Keane, but it remains importantas a source of quantitative estimates.

Under the title “ant in a labyrinth,” de Gennesproposed the use of a random walk to explore thegeometry of an open cluster. In a beautiful piece of work

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[14], Harry showed the existence of a measure known asthe “incipient infinite cluster,” obtained effectively by con-ditioning on critical percolation possessing an infinite clus-ter at the origin. He then proved that random walk 𝑋𝑛 onthis cluster is subdiffusive in that there exists 𝜖 > 0 such

that 𝑋𝑛/𝑛12−𝜖 → 0. This is in contrast to the situation on

ℤ𝑑 for which 𝑋𝑛 has order 𝑛12 . This “slowing down” oc-

curs because the walk spends time in blind alleys of theincipient infinite cluster.

Whereas it was not possible to obtain exact results forcritical percolation, Harry gave a precise solution to thecorresponding problem on the family tree 𝑇 of a criticalbranching process, conditioned on non-extinction. Herealso there are blind alleys, but it is possible to estimate thetime spent in them. It turns out that the displacement 𝑋𝑛of the walk has order 𝑛

13 and, moreover, Harry computed

the limit distribution of 𝑋𝑛/𝑛13 .

In joint work with Grimmett and Zhang on supercriticalpercolation, he showed that random walk on the infinitecluster in 𝑑 ≥ 2 dimensions is recurrent if and only if 𝑑 = 2.This basic result stimulated others to derive precise heatkernel estimates for random walks on percolation clustersand other random networks.

We mention one further result for classical percolationon ℤ𝑑. When 𝑝 < 𝑝c, the tail of |𝐶| decays exponentially to0, in that ℙ𝑝(|𝐶| = 𝑛) ≤ 𝑒−𝛼𝑛 for some 𝛼(𝑝) > 0. Mattersare more complicated when 𝑝 > 𝑝c, since large clusters“prefer” to be infinite. It turns out that the interior of alarge cluster (of size 𝑛, say) typically resembles that of theinfinite cluster, and its finiteness is controlled by its bound-ary (of order 𝑛(𝑑−1)/𝑑). As a result, |𝐶| should have a tailof order exp{−𝑐𝑛(𝑑−1)/𝑑}. There was a proof by Aizenman,Delyon, and Souillard that

ℙ𝑝(|𝐶| = 𝑛) ≥ exp{−𝛽𝑛(𝑑−1)/𝑑}for 𝛽(𝑝) > 0. Kesten and Zhang showed, by a block argu-ment, the complementary inequality

ℙ𝑝(|𝐶| = 𝑛) ≤ exp{−𝛾𝑛(𝑑−1)/𝑑}for some 𝛾(𝑝) > 0. When 𝑑 ≥ 3, they were in fact onlyable to show this for 𝑝 exceeding a certain value 𝑝slab, butthe conclusion for 𝑝 > 𝑝c followed once Grimmett andMarstrand had proved the slab limit 𝑝c = 𝑝slab. Sharpasymptotics were established later by Alexander, Chayes,and Chayes when 𝑑 = 2, and by Cerf in the more challeng-ing situation of 𝑑 = 3, in their work on the Wulff construc-tion for percolation.

First-passage percolation was introduced by Hammers-ley and Welsh in 1965 as an extension of classical percola-tion in which each edge has a random “passage time,” andone studies the set of vertices reached from the origin along

Figure 9. Harry Kesten with John Hammersley, Oxford, 1993.

paths of length not exceeding a given value. This is wherethe notion of a subadditive stochastic process was intro-duced, and an ergodic theorem first proved. The theory ofsubadditivity was useful throughout Harry’s work on dis-ordered networks, and indeed he noted that it provided an“elegant” proof of his 1960 theorem with Furstenberg onproducts of random matrices.

Harry turned towards first-passage percolation around1979, and he resolved a number of open problems, andposed others, in a series of papers spanning nearly tenyears. He established fundamental properties of the timeconstant, including positivity under a natural condition,and continuity as a function of the underlying distribution(with Cox), together with a large deviation theorem forpassage times (with Grimmett). Perhaps his most notablecontribution was a theory of duality in three dimensionsakin to Whitney duality of two dimensions, as expoundedin his Saint-Flour notes [8]. The dual process is upon pla-quettes, and dual surfaces occupy the role of dual paths intwo dimensions. This leads to some tricky geometrical is-sues concerning the combinatorics and topology of dualsurfaces which have been largely answered since by Zhang,Rossignol, Theret, Cerf, and others.

Percolation and its cousins provide the environment fora number of related processes to which Harry contributedsubstantial results. He was an enthusiastic contributor totoo many collaborative ventures to be described in fullhere. As a sample wemention word percolation (with Ben-jamini, Sidoravicius, Zhang), 𝜌-percolation (with Su), ran-dom lattice animals (with Cox, Gandolfi, Griffin), and uni-form spanning trees (with Benjamini, Peres, Schramm).

PostscriptThis memoir describes only a sample of Harry Kesten’simpact within the probability and statistical physics com-munities. Those seeking more may read Rick Durrett’saccount [2] or the recent article [7]. The latter includes

830 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6

summaries of Harry’s work in other areas, such as quasi-stationary distributions of Markov chains and bounded re-mainder sets in Euclidean dynamics.

ACKNOWLEDGMENTS. The authors are grateful toseveral people for their advice: Michael Kesten for ac-cess to the Kesten family papers and for his personalreminiscences, Laurent Saloff-Coste for biographicalmaterial from Cornell, Frank den Hollander for adviceon Dutch matters, Rob van den Berg for his suggestions,Hillel Furstenberg for his comments on a draft, andthree anonymous readers for their suggestions.

References[1] Emmanuel Breuillard, Expander graphs, property (𝜏) and

approximate groups, Geometric group theory, IAS/Park CityMath. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014,pp. 325–377. MR3329732

[2] R. Durrett,Harry Kesten’s publications: a personal perspective,Perplexing Problems in Probability, Birkhäuser Boston,1999, pp. 1–33. MR1703122

[3] E. B. Dynkin, Interview with Harry Kesten, Collectionof mathematics interviews, 1995. dynkincollection.library.cornell.edu/biographies/866.

[4] A. Elon, The Pity of It All: A Portrait of the German-JewishEpoch, 1743–1933, Picador, New York, NY, 2003.

[5] Natanel (Sonyu) Farber, Podwołocyska, my town, Pod-wołoczyska and its Surroundings, 1988. https://www.jewishgen.org/yizkor/podvolochisk/pod001.html.

[6] H. Furstenberg and H. Kesten, Products of random matrices,Ann. Math. Statist. 31 (1960), 457–469. MR121828

[7] G. R. Grimmett, Harry Kesten’s work in probability theory(2020). To appear.

[8] G. R. Grimmett and H. Kesten, Percolation Theory atSaint-Flour, Probability at Saint-Flour, Springer, Heidel-berg, 2012. MR2905932

[9] H. Kesten, On the number of self-avoiding walks, J. Mathe-matical Phys. 4 (1963), 960–969. MR152026

[10] H. Kesten, Hitting probabilities of single points for pro-cesses with stationary independent increments, Memoirs of theAmerican Mathematical Society, No. 93, American Mathe-matical Society, Providence, R.I., 1969. MR0272059

[11] H. Kesten, Random difference equations and renewal theoryfor products of random matrices, Acta Math. 131 (1973), 207–248, DOI 10.1007/BF02392040. MR440724

[12] H. Kesten, The critical probability of bond percolation on thesquare lattice equals

12, Comm. Math. Phys. 74 (1980), 41–

59. MR575895[13] H. Kesten, Percolation Theory for Mathematicians,

Birkhäuser, Boston, 1982. Available at pi.math.cornell.edu/~kesten/kesten-book.html. MR692943

[14] H. Kesten, Subdiffusive behavior of random walk on arandom cluster, Ann. Inst. H. Poincare Probab. Statist. 22(1986), no. 4, 425–487. MR871905

[15] H. Kesten, How long are the arms in DLA?, J. Phys. A 20(1987), no. 1, L29–L33. MR873177

[16] H. Kesten, Scaling relations for 2D-percolation, Comm.Math. Phys. 109 (1987), no. 1, 109–156. MR879034

[17] H. Kesten, Relations between solutions to a discrete and con-tinuous Dirichlet problem, Random Walks, Brownian Mo-tion, and Interacting Particle Systems, Birkhäuser Boston,1991, pp. 309–321. MR1146455

[18] H. Kesten, Some highlights of percolation, Proceed-ings of the International Congress of Mathematicians2002, Higher Ed. Press, Beijing, 2002. Video availableat https://www.mathunion.org/icm/icm-videos/icm-2002-videos-beijing-china/icm-beijing-videos-27082002. MR1989192

[19] H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law forrandom walk in a random environment, Compositio Math.30 (1975), 145–168. MR380998

[20] H. Kesten and J. Th. Runnenburg, Priority in waiting lineproblems. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 60 =Indag. Math. 19 (1957), 312–324, 325–336. MR89775,MR89776

[21] H. Kesten and B. P. Stigum, A limit theorem for multidi-mensional Galton–Watson processes, Ann. Math. Statist. 37(1966), 1211–1223. MR198552

[22] G. F. Lawler, O. Schramm, and W. Werner, On the scal-ing limit of planar self-avoiding walk, Fractal Geometry andApplications: a Jubilee of Benoît Mandelbrot, Part 2, 2004,pp. 339–364. MR2112127

Geoffrey R.Grimmett

Gregory F. Lawler

Credits

Figure 1 is courtesy of Cornell University.Figure 2 is courtesy of Rob van den Berg.Figure 3 is courtesy of Hugo Grimmett.Figure 4 is by Konrad Jacobs. Source: Archives of the Mathe-

matisches Forschungsinstitut Oberwolfach.Figure 5 is courtesy of Michael A. Morgan.Figures 6–9 are courtesy of Geoffrey Grimmett.Photo of Geoffrey R. Grimmett is courtesy of Downing Col-

lege, Cambridge.Photo of Gregory F. Lawler is courtesy of Jean Lachat/

University of Chicago.

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