# Harry Kesten

Harry Kesten
Harry Kesten at Cornell University, 1970
Born Harry Kesten
19 November 1931 (age 85)
Germany
Nationality American
Fields
Institutions
Alma mater
Thesis Symmetric Random Walks on Groups (1958)
Doctoral students
• Barry Belkin
• Maury Bramson
• Pablo Fierens
• Edmond Granirer
• James Hutton
• Kenji Ichihara
• Antal Jarai
• Steven Kalikow
• Eric Key
• Sungchul Lee
• J.-H. Lou
• Heinrich Matzinger
• Grant Ritter
• Rahul Roy
• Tom Ryan
• David Stephenson
• David Tanny
• Yu Zhang[1]
Notable awards
Spouse Doraline Kesten
Children Michael Kesten
Website
www.math.cornell.edu/People/Faculty/kesten.html

Harry Kesten (born 19 November 1931, Germany) is an American mathematician best known for his work in probability, most notably on random walks on groups and graphs, random matrices, branching processes, and percolation theory.

## Biography

Kesten grew up in the Netherlands, where he moved with his parents in 1933 to escape the Nazis. He received his Ph.D. in 1958 at Cornell University under supervision of Mark Kac. He was an instructor at Princeton University and the Hebrew University before returning to Cornell where he is now Professor Emeritus of mathematics.

## Mathematical work

Kesten's work includes many fundamental contributions across almost the whole of probability,[6] including the following highlights.

• Random walks on groups. In his 1958 PhD thesis, Kesten studied symmetric random walks on countable groups G generated by a jump distribution with support G. He showed that the spectral radius equals the exponential decay rate of the return probabilities.[7] He showed later that this is strictly less than 1 if and only if the group is non-amenable.[8] The last result is known as Kesten's criterion for amenability. He calculated the spectral radius of the d-regular tree, namely ${\displaystyle 2{\sqrt {d-1}}}$.
• Products of random matrices. Let ${\displaystyle Y_{n}=X_{1}X_{2}\dots X_{n}}$ be the product of the first n elements of an ergodic stationary sequence of random ${\displaystyle k\times k}$ matrices. With Furstenberg in 1960, Kesten showed the convergence of ${\displaystyle n^{-1}\log ^{+}\|Y_{n}\|}$, under the condition ${\displaystyle E(\log ^{+}\|X_{1}\|)<\infty }$.[9]
• Self-avoiding walks. Kesten's ratio limit theorem states that the number ${\displaystyle \sigma _{n}}$ of n-step self-avoiding walks from the origin on the integer lattice satisfies ${\displaystyle \sigma _{n+2}/\sigma _{n}\to \mu ^{2}}$ where ${\displaystyle \mu }$ is the connective constant. This result remains unimproved despite much effort.[10] In his proof, Kesten proved his pattern theorem, which states that, for a proper internal pattern P, there exists ${\displaystyle \alpha }$ such that the proportion of walks containing fewer than ${\displaystyle \alpha n}$ copies of P is exponentially smaller than ${\displaystyle \sigma _{n}}$.[11]
• Branching processes. Kesten and Stigum showed that the correct condition for the convergence of the population size, normalized by its mean, is that ${\displaystyle E(L\log ^{+}L)<\infty }$ where L is a typical family size.[12] With Ney and Spitzer, Kesten found the minimal conditions for the asymptotic distributional properties of a critical branching process, as discovered earlier, but subject to stronger assumptions, by Kolmogorov and Yaglom.[13]
• Random walk in a random environment. With Kozlov and Spitzer, Kesten proved a deep theorem about random walk in a one-dimensional random environment. They established the limit laws for the walk across the variety of situations that can arise within the environment.[14]
• Diophantine approximation. In 1966, Kesten resolved a conjecture of Erdős and Szűsz on the discrepancy of irrational rotations. He studied the discrepancy between the number of rotations by ${\displaystyle \xi }$ hitting a given interval I, and the length of I, and proved this bounded if and only if the length of I is a multiple of ${\displaystyle \xi }$.[15]
• Diffusion-limited aggregation. Kesten proved that the growth rate of the arms in d dimensions can be no larger than ${\displaystyle n^{2/(d+1)}}$.[16][17]
• Percolation. Kesten's most famous work in this area is his proof that the critical probability of bond percolation on the square lattice equals 1/2.[18] He followed this with a systematic study of percolation in two dimensions, reported in his book Percolation Theory for Mathematicians.[19] His work on scaling theory and scaling relations [20] has since proved key to the relationship between critical percolation and Schramm-Loewner evolution.[21]
• First passage percolation. Kesten's results for this growth model are largely summarized in Aspects of First Passage Percolation.[22] He studied the rate of convergence to the time constant, and contributed to the topics of subadditive stochastic processes and concentration of measure. He developed the problem of maximum flow through a medium subject to random capacities.

A volume of papers was published in Kesten's honor in 1999.[23]

## References

1. ^ a b
2. ^ List of Wald Lecturers
3. ^
4. ^ "H. Kesten". Royal Netherlands Academy of Arts and Sciences. Retrieved 17 July 2015.
5. ^ List of Fellows of the American Mathematical Society, retrieved 2013-01-27.
6. ^ Durrett, R., Harry Kesten's publications: a personal perspective. Perplexing problems in probability, 1–33, Progr. Probab., 44, Birkhäuser, Boston MA, 1999.
7. ^ Kesten, H., Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959), 336--354.
8. ^ Kesten, H., Full Banach mean values on countable groups. Math. Scand. 7 (1959), 146–156.
9. ^ Furstenberg, H. and Kesten, H., Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469.
10. ^ Madras, N. and Slade, G., The self-avoiding walk, Birkhäuser, Boston, 1993.
11. ^ Kesten, H., On the number of self-avoiding walks. I and II. J. Mathematical Phys. 4 (1963) 960--969, 5 (1964), 1128--1137.
12. ^ Kesten, H. and Stigum, B, A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Statist. 37 (1966), 1211–1223.
13. ^ Kesten, H., Ney, P. and Spitzer, F., The Galton-Watson process with mean one and finite variance, Theory Probab. Appl. 11 (1966), 513–540.
14. ^ Kesten, H., Kozlov, M. V., Spitzer, F. A limit law for random walk in a random environment. Compositio Math. 30 (1975), 145–168.
15. ^ Kesten, H. (1966). "On a conjecture of Erdős and Szüsz related to uniform distribution mod 1". Acta Arith. 12: 193–212.
16. ^ Kesten, H., How long are the arms in DLA? J. Phys. A 20 (1987), L29--L33.
17. ^ Kesten, H., Upper bounds for the growth rate of DLA, Physica A 168 (1990), 529-535.
18. ^ Kesten, H., The critical probability of bond percolation on the square lattice equals 1/2. Comm. Math. Phys. 74 (1980), 41--59.
19. ^ Kesten, H. (1982), Percolation Theory for Mathematicians.
20. ^ Kesten, H., Scaling relations for 2D-percolation. Comm. Math. Phys. 109 (1987), 109--156.
21. ^ Smirnov, S., Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 239–244.
22. ^ Kesten, H., Aspects of First Passage Percolation. École d'été de probabilités de Saint-Flour, XIV—1984, 125–264, Lecture Notes in Math., 1180, Springer, Berlin, 1986.
23. ^ Perplexing problems in probability: Festschrift in honor of Harry Kesten, Bramson, M. and Durrett, R., eds, Progr. Probab., 44, Birkhäuser, Boston MA, 1999.
24. ^ Wierman, John (1984). "Review: Percolation theory for mathematicians, by Harry Kesten" (PDF). Bull. Amer. Math. Soc. (N.S.). 11 (2): 404–409. doi:10.1090/s0273-0979-1984-15331-x.