Harry Kesten at Cornell in 1970 and in his later years

On March 29, 2019 Harry Kesten lost a decade-long battle with Parkinson’s disease. His passing is a sad event, so I would like to find solace in celebrating his extraordinary career. In addition I hope you will learn a little more about his work by reading this.

Harry was born in Duisburg Germany on November 19, 1931. His parents escaped from the Nazis in 1933 and moved to Amsterdam. After studying in Amsterdam, he was a research assistant at the Mathematical Center there until 1956 when he came to Cornell. He received his Ph.D. in 1958 at Cornell University under supervision of Mark Kac.

In his 1958 thesis on *Symmetric Random Walks*, he showed that the spectral radius equals the exponential decay rate of the return to 0, and the latter is strictly less than 1 if and only if the group is non-amenable This work has been cited 206 times and is his second most cited publication (according to MathSciNet). Harry was an instructor at Princeton University for one year and at the Hebrew University for two years before returning to Cornell, where he spent the rest of his career. While in Israel, he and Furstenberg wrote their classic paper on *Products of Random Matrices*.

In the 1960s, he wrote a number of papers that proved sharp or very general results on random walks, branching process, etc. One of the most famous of these is the 1966 Kesten-Stigum theorem which shows that a a normalized branching process Z_{n}/μ^{n} has a nontrival limit if and only if the offspring distribution has E(X log^{+} X) < ∞. In 1966 he also proved a conjecture of Erdös and Szuzu about the discrepancy between the number of rotations of a point on the unit circle hitting an interval and its length. Foreshadowing his work in physics, he showed in 1963 that the number of self-avoiding walks of length n satisfied σ_{n+2}/σ_{n} → μ^{2} , where μ is the connective constant.

Harry’s almost 200 papers have been cited 3781 times by 2329 authors However, these statistics underestimate his impact. In baseball terms, Harry was a closer. When he wrote a paper about a topic, his results often eliminated the need for future work on it. One of Harry’s biggest weaknesses is that he was too smart. When most of us are confronted with a problem, we need to try different approaches to find a route through the woods to a solution. Harry simply got on his bulldozer and drove over all obstacles. He needed 129 pages in the Memoirs of the AMS to answer the question: “Which processes with stationary independent increments hit points?”, a topic he spoke about at the International Congress at Nice in 1970.

In 1984 Harry gave lectures on first passage percolation at the St. Flour Probability Summer School. This subject dates back to Hammersley’s 1966 paper and was greatly advanced by Smythe and Weirman’s 1978 book. However, Harry’s paper attracted a number of people to work on the subject and it has continued to be a very active area. See *50 years of First Passage Percolation* by Auffinger, Damron, and Hanson for more details. You can buy this book from the AMS or download it from the arXiv. I find it interesting that Harry lists only six papers on his Cornell web page. Five have already been mentioned. The sixth is *On the speed of convergence in first-passage percolation,* Ann. Appl. Probab. 3(1993), 296–338.

Harry has worked in a large number of areas. There is not enough space for a systematic treatment so I will just tease you with a list of titles. Sums of stationary sequences cannot grow slower than linearly. Random difference equations and renewal theory for products of random matrices. Subdiffusive behavior of a random walk on a random cluster. Greedy lattice animals. How long are the arms of DLA? If you want to try to solve a problem Harry couldn’t, look at his papers on Diffusion Limited Aggregation.

In the late 1990s, Maury Bramson and I organized a conference in honor of Harry’s 66 2/3’s birthday. (We missed 65 and didn’t want to wait for 70.) A distinguished collection of researchers gave talks and many contributed to a volume of papers in his honor called *Perplexing Problems in Probability*. The 21 papers in the volume provide an interesting snapshot of research at the time. If you want to know more about Harry’s first 150 papers, you can read my 32 page summary of his work that appears in that volume.

According to math genealogy, Harry supervised 17 Cornell Ph.D. students who received their degrees between 1962-2003. Maury Bramson and Steve Kalikow were part of the Cornell class of 1977 that included Larry Gray and David Griffeath who worked with Frank Spitzer. (Fortunately, I graduated in 1976!). Yu Zhang followed in Harry’s footsteps and made a number of contributions to percolation and first passage percolation. I’ll let you use google to find out about the work Kenji Ichihara, Antal Jarai, Sungchul Lee, Henry Matzinger, and David Tandy.

Another ‘broader impact” of Harry’s work came from his collaborations with a long list of distinguished co-authors: Vladas Sidorovicius (12 papers), Ross Maller (10) , Frank Spitzer (8), Geoffrey Grimmett (7), Yu Zhang (7), Itai Benjamini (6), J.T. Runnenberg (5), Roberto Schonmann (4), Rob van den Berg (4), … I wrote 4 papers with him, all of which were catalyzed by an interaction with another person. In response to a question asked by Larry Shepp, we wrote a paper about an inhomogeneous percolation which was a precursor to work by Bollobas, Janson, and Riordan. *Making money from fair games*, joint work with Harry and Greg Lawler, arose from a letter A. Spataru wrote to Frank Spitzer. I left it to Harry and Greg to sort out the necessary conditions.

Harry wrote 3 papers with two very different versions of Jennifer Chayes. With a leather-jacketed Cornell postdoc, her husband Lincoln Chayes, Geoff Grimmett and Roberto Schonmann, he studied “The correlation length for the high density phase.” With the manager of the Microsoft research group, her husband Christian Borgs, and Joel Spencer he wrote two papers, one on the birth of the infinite component in percolation and another on conditions implying hyperscaling.

As you might guess from my narrative, Kesten received a number of honors. He won the Brouwer medal in 1981. Named after L.E.J. Brouwer it is The Netherlands’ most prestigious award in mathematics. In 1983 he was elected to the National Academy of Science. In 1986 he gave the IMS’ Wald Lectures. In 1994 he won the Polya Prize from SIAM. In 2001 he won the AMS’ Steele Prize for lifetime achievement.

Being a devout orthodox Jew, Harry never worked on the Sabbath. On Saturdays in Ithaca, I would often drive past him taking a long walk on the aptly named Freese Road, lost in thought. Sadly Harry is now gone, but his influence on the subject of probability will not be forgotten.