Kesten's work includes many fundamental contributions across almost the whole of probability,[6][8][9] including the following highlights.
Random walks ongroups. In his 1958 PhD thesis, Kesten studied symmetric random walks on countable groupsG generated by a jump distribution with supportG. He showed that the spectral radius equals the exponential decay rate of the return probabilities.[10] He showed later that this is strictly less than 1 if and only if the group isnon-amenable.[11] The last result is known asKesten's criterion for amenability. He calculated the spectral radius of thed-regular tree, namely.
Products ofrandom matrices. Let be the product of the firstn elements of an ergodic stationary sequence of random matrices. WithFurstenberg in 1960, Kesten showed the convergence of, under the condition.[12]
Self-avoiding walks. Kesten's ratio limit theorem states that the number ofn-step self-avoiding walks from the origin on the integer lattice satisfies where is theconnective constant. This result remains unimproved despite much effort.[13] In his proof, Kesten proved his pattern theorem, which states that, for a proper internal patternP, there exists such that the proportion of walks containing fewer than copies ofP is exponentially smaller than.[14]
Branching processes. Kesten and Stigum showed that the correct condition for the convergence of the population size, normalized by its mean, is that whereL is a typical family size.[15] With Ney andSpitzer, Kesten found the minimal conditions for the asymptotic distributional properties of a critical branching process, as discovered earlier, but subject to stronger assumptions, byKolmogorov andYaglom.[16]
Random walk in a random environment. With Kozlov andSpitzer, 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.[17]
Diophantine approximation. In 1966, Kesten resolved a conjecture ofErdős and Szűsz on the discrepancy of irrational rotations. He studied the discrepancy between the number of rotations by hitting a given intervalI, and the length ofI, and proved this bounded if and only if the length ofI is a multiple of.[18]
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.[21] He followed this with a systematic study of percolation in two dimensions, reported in his bookPercolation Theory for Mathematicians.[22] His work on scaling theory and scaling relations[23] has since proved key to the relationship between critical percolation andSchramm–Loewner evolution.[24]
First passage percolation. Kesten's results for this growth model are largely summarized inAspects of First Passage Percolation.[25] He studied the rate of convergence to the time constant, and contributed to the topics ofsubadditive stochastic processes andconcentration of measure. He developed the problem ofmaximum flow through a medium subject to random capacities.
A volume of papers was published in Kesten's honor in 1999.[26] The Kesten memorial volume ofProbability Theory and Related Fields[27] contains a full list of the dedicatee's publications.
Kesten, Harry (1969). "A convolution equation and hitting probabilities of single points for processes with stationary independent increments".Bull. Amer. Math. Soc.75 (3):573–578.doi:10.1090/s0002-9904-1969-12245-7 (inactive September 13, 2025).MR0251797.{{cite journal}}: CS1 maint: DOI inactive as of September 2025 (link)
^Durrett, R., Harry Kesten's publications: a personal perspective. Perplexing problems in probability, 1–33, Progr. Probab., 44, Birkhäuser, Boston MA, 1999.
^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.
^Perplexing problems in probability: Festschrift in honor of Harry Kesten, Bramson, M. and Durrett, R., eds, Progr. Probab., 44, Birkhäuser, Boston MA, 1999
^H. Duminil-Copin, G. R. Grimmett, ed. (2021). "Special issue in honor of the life and work of Harry Kesten".Probability Theory and Related Fields.181:1–756.
