Regularity of self-diffusion coefficient. (English) Zbl 0822.60089

Freidlin, Mark I. (ed.), The Dynkin Festschrift. Markov processes and their applications. In celebration of Eugene B. Dynkin’s 70th birthday. Boston, MA: Birkhäuser. Prog. Probab. 34, 387-397 (1994).
Some symmetric random walks with simple exclusion are investigated. They are Markov processes on the configuration space \(\{0,1\}^{\mathbb{Z}^ d}\) with invariant measures being the Bernoulli product measures \(P_ \rho\) with the density \(\rho \in (0,1)\) of particles. Former results show that under suitable scaling the process describing the motion of a tagged particle starting from the origin with \(P_ \rho\)-distribution of the other particles converges to the Brownian motion with covariance matrix (called self-diffusion matrix in this context) \(S(\rho)\) described by a variational formula. The behaviour of \(S(\rho)\) is studied. Some bounds are derived for all \(d \geq 1\) (nearest neighbour walk for \(d = 1\) is excluded) using the variational formula and a suitable coupling. Lipschitz property of \(S(\rho)\), \(\rho \in [0,1]\), is shown for \(d \geq 3\) and it is based on a lemma showing, for \(d \geq 3\), that for \(f \in l_ 2(\mathbb{Z}^ d)\) and \(x_ 0 \in \mathbb{Z}^ d\) there is a connected path \(\{x_ k \mid k = 0,1,\dots \}\) with \(\| x_ k\| \to \infty\) such that \(\sum^ \infty_{k = 0} | f(x_ k)| \leq \text{const}(d) \cdot \| f\|_ 2\). An asymmetric random walk is used to prove it.
For the entire collection see [Zbl 0808.00010].
Reviewer: P.Holicky (Praha)


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
60G50 Sums of independent random variables; random walks