Stabilization of stochastic networks in Markovian environment
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918411216355328 |
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| author | Kaiser, Robin Klötzer, Martin Sava-Huss, Ecaterina |
| author_facet | Kaiser, Robin Klötzer, Martin Sava-Huss, Ecaterina |
| contents | We establish criteria under which stochastic networks in a Markovian environment stabilize, thus confirming Conjecture 7.2 from Levine-Greco [GL23]. The networks evolve on finite connected graphs $G=(V,E)$, and their dynamics are encoded by $V \times V$ toppling matrices $M$, whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter $ρ$ which depends on the largest eigenvalue of the matrix $M+αI$, with $α=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25606 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stabilization of stochastic networks in Markovian environment Kaiser, Robin Klötzer, Martin Sava-Huss, Ecaterina Probability 60J80, 60F05, 60F15 We establish criteria under which stochastic networks in a Markovian environment stabilize, thus confirming Conjecture 7.2 from Levine-Greco [GL23]. The networks evolve on finite connected graphs $G=(V,E)$, and their dynamics are encoded by $V \times V$ toppling matrices $M$, whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter $ρ$ which depends on the largest eigenvalue of the matrix $M+αI$, with $α=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$. |
| title | Stabilization of stochastic networks in Markovian environment |
| topic | Probability 60J80, 60F05, 60F15 |
| url | https://arxiv.org/abs/2603.25606 |