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 .
Products of random matrices. Let be the product of the first n elements of an ergodic stationary sequence of random matrices. With Furstenberg in 1960, Kesten showed the convergence of, under the condition.
Self-avoiding walks. Kesten's ratio limit theorem states that the number of n-step self-avoiding walks from the origin on the integer lattice satisfies where 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 such that the proportion of walks containing fewer than copies of P is exponentially smaller than.
Branching processes. Kesten and Stigum showed that the correct condition for the convergence of the population size, normalized by its mean, is that 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 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.
Diffusion-limited aggregation. Kesten proved that the growth rate of the arms in d dimensions can be no larger than.
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.
