Abstract
This paper is concerned with the contact process with random vertex weights on regular trees, and studies the asymptotic behavior of the critical infection rate as the degree of the trees increasing to infinity. In this model, the infection propagates through the edge connecting vertices x and y at rate λρ(x)ρ(y) for some λ > 0, where {ρ(x), x ∈ T d} are independent and identically distributed (i.i.d.) vertex weights. We show that when d is large enough, there is a phase transition at λc(d) ∈ (0,∞) such that for λ < λc(d), the contact process dies out, and for λ > λc(d), the contact process survives with a positive probability. Moreover, we also show that there is another phase transition at λe(d) such that for λ < λe(d), the contact process dies out at an exponential rate. Finally, we show that these two critical values have the same asymptotic behavior as d increases.
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Acknowledgements
The authors are grateful to the anonymous referees for their detailed comments and suggestions. Pan and Chen are grateful to the financial support from the National Natural Science Foundation of China (Grant Nos. 11371040, 11531001). Xue is grateful to the financial support from the National Natural Science Foundation of China (Grant No. 11501542) and the financial support from Beijing Jiaotong University (Grant No. KSRC16006536).
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Pan, Y., Chen, D. & Xue, X. Contact process on regular tree with random vertex weights. Front. Math. China 12, 1163–1181 (2017). https://doi.org/10.1007/s11464-017-0633-4
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DOI: https://doi.org/10.1007/s11464-017-0633-4