Harry Kesten

Random difference equations and Renewal theory for products of random matrices

where Mn and Qn are random d • d matrices respectively d-vectors and Yn also is a d-vector. Throughout we take the sequence of pairs (Mn, Q~), n >/1, independently and identically distributed. The equation (1.1) arises in various contexts. We first met a special case in a paper by Solomon, [20] sect. 4, which studies random walks in random environments. Closely related is the fact tha t if Yn(i) is the expected number of particles of type i in the nth generation of a d-type branching process in a random environment with immigration, then Yn = (Yn(1) ..... Yn(d)) satisfies (1.1) (Qn represents the immigrants in the nth generation). (1.1) has been used for the amount of radioactive material in a compar tment ([17]) and in control theory [9 a]. Moreover, it is the principal feacture in a model for evolution and cultural inheritance by Cavalli-Sforza and Feldman [2]. Notice also tha t the dth order linear difference equation

The Critical Probability of Bond Percolation on the Square Lattice Equals 1/2*

We prove the statement in the title of the paper.

A limit theorem related to a new class of self similar processes

SummaryWe study partial sums of a stationary sequence of dependent random variables of the form

Scaling relations for 2D-percolation

W_n = \sum\limits_1^n {\xi \left( {S_k } \right)}

A Limit Theorem for Turbulent Diffusion

. Here Sk=X1 + ... +Xk where the Xiare i.i.d. integer valued, and ξ(n), n∈ℤ are also i.i.d. and independent of the Xs. It is assumed that the Xs and ξs belong to the domains of attraction of different stable laws of indices 1

The incipient infinite cluster in two-dimensional percolation

\frac{1}{2}