Andre Toom

TAILS IN HARNESSES

AbstractWe consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesast, wheres ∈ Zd andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Zd+ define Δσa′s as follows. If σ=(0,...,0), σ=(0,...,0), Δσast=ast. Then by induction,

How I teach word problems

\Delta _{\sigma + e_1 } a_s^t = \Delta _\sigma a_{s + e_1 }^t - \Delta _\sigma a_s^t

Variable-length analog of Stavskaya process: A new example of misleading simulation

whereei is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δσast diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δσast diverges; if P-decay(v)>(d+2)/(d+|σ|), Δσast, converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|r) is finite. Let E-decay(v)>0. Whenever Δσast converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(limast)=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δσast)=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δσast)=E-decay(ν).

On Large Isolated Regions in Supercritical Percolation

We complete the proof of a necessary and sufficient condition for existence of non-trivial critical values for some classes of random processes with local interaction, where the space is a real plane. Our operators are superpositions of a deterministic operator and a one-sided random noise, where the noise is standard and the geometric properties of the deterministic operator are crucial.

Stochastic cellular systems : ergodicity, memory, morphogenesis

ABSTRACT This article tells how the author teaches university freshmen to solve word problems and how it serves their intellectual development.

Mathematical Circles: Russian Experience

We prove algorithmical unsolvability of the ergodicity problem for a class of one-dimensional translation-invariant random processes with local interaction with continuous time, also known as interacting particle systems. The set of states of every component is finite, the interaction occurs only between nearest neighbors, only one particle can change its state at a time and all rates are 0 or 1.

Locally Interacting Systems and Their Application in Biology

This article presents a new example intended to showcase limitations of computer simulations in the study of random processes with local interaction. For this purpose, we examine a new version of the well-known Stavskaya process, which is a discrete-time analog of the well-known contact processes. Like the bulk of random processes studied till now, the Stavskaya process is constant-length, that is, its components do not appear or disappear in the course of its functioning. The process, which we study here and call Variable Stavskaya, VS, is similar to Stavskaya; it is discrete-time; its states are bi-infinite sequences, whose terms take only two values (denoted here as “minus” and “plus”), and the measure concentrated in the configuration “all pluses” is invariant. However, it is a variable length, which means that its components, also called particles, may appear and disappear under its action. The operator VS is a composition of the following two operators. The first operator, called “birth,” depends on...

Non-ergodicity in a 1-D particle process with variable length

AbstractWe consider supercritical vertex percolation in