Broadcasting on trees and the Ising model.(English)Zbl 1052.60076

The authors consider the following process. At the root $$\rho$$ of a tree $$T$$ there is a binary random variable (spin) taking its values with equal probabilities. This bit is then propagated through the tree as follows: each vertex receives the bit from its parent unchanged with probability $$(1-\varepsilon)$$ and changed with probability $$\varepsilon\in (0, 1/2]$$. The events on different edges are statistically independent. Let $$W$$ be a subset of the set of vertices of $$T$$. For such $$W$$, let the set of the bits arrived at each element of $$W$$ be given. Then one tries to reconstruct the original bit on the base of this information. Clearly the probability to do this correctly is at least $$1/2$$; denote this probability by $$(1+\Delta)/2$$, where $$\Delta = \Delta (T, W, \varepsilon)$$. The main results of the paper are upper and lower bounds for such $$\Delta(T, W, \varepsilon)$$. If the tree is infinite, the bounds yield the following: $$\lim_{n\rightarrow +\infty} \Delta(T, T_n \varepsilon) > 0$$ if $$\varepsilon < \varepsilon_c$$ and $$\lim_{n\rightarrow +\infty} \Delta(T, T_n \varepsilon) = 0$$ if $$\varepsilon > \varepsilon_c$$. Here $$T_n$$ is the set of the vertices of $$T$$ of $$n$$th level. For regular trees, the problem of determining $$\varepsilon_c$$ was solved by P. M. Bleher, J. Ruiz and V. A. Zagrebnov [J. Stat. Phys. 79, 473–482 (1995; Zbl 1081.82515)].

MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 68M10 Network design and communication in computer systems 68R99 Discrete mathematics in relation to computer science 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Zbl 1081.82515
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