Kesten's work includes many fundamental contributions across almost the whole of probability, 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. He showed later that this is strictly less than 1 if and only if the group is
non-amenable. The last result is known as ''Kesten's criterion for amenability
. He calculated the spectral radius of the d''-regular tree, namely \frac{2\sqrt{d-1}}{d}. •
Products of random matrices. Let Y_n=X_1 X_2\cdots X_n be the product of the first
n elements of an ergodic stationary sequence of random k \times k matrices. With
Furstenberg in 1960, Kesten showed the convergence of n^{-1}\log^+\|Y_n\|, under the condition E (\log^+\|X_1\|). •
Self-avoiding walks. Kesten's ratio limit theorem states that the number \sigma_n of
n-step self-avoiding walks from the origin on the integer lattice satisfies \sigma_{n+2}/\sigma_n \to \mu^2 where \mu is the
connective constant. This result remains unimproved despite much effort. In his proof, Kesten proved his pattern theorem, which states that, for a proper internal pattern
P, there exists \alpha such that the proportion of walks containing fewer than \alpha n copies of
P is exponentially smaller than \sigma_n. •
Branching processes. Kesten and Stigum showed that the correct condition for the convergence of the population size, normalized by its mean, is that E(L\log^+ L) where
L is a typical family size. 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. •
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. •
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 \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 \xi. •
Diffusion-limited aggregation. Kesten proved that the growth rate of the arms in
d dimensions can be no larger than n^{2/(d+1)}. •
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. He followed this with a systematic study of percolation in two dimensions, reported in his book
Percolation Theory for Mathematicians. His work on scaling theory and scaling relations has since proved key to the relationship between critical percolation and
Schramm–Loewner evolution. •
First passage percolation. Kesten's results for this growth model are largely summarized in
Aspects of First Passage Percolation. 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. The Kesten memorial volume of
Probability Theory and Related Fields contains a full list of the dedicatee's publications. and
Roland Dobrushin in
Oxford, 1993 ==Selected works==