Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914436493606912 |
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| author | Morfe, Peter S. |
| author_facet | Morfe, Peter S. |
| contents | This paper analyzes a class of recursive distributional equations (RDE's) proposed by Gurel-Gurevich [17] and involving a bias parameter $p$, which includes the logarithm of the resistance of the series-parallel graph. A discrete-time evolution equation resembling a quasilinear Fisher-KPP equation is derived to describe the CDF's of solutions. When the bias parameter $p = \frac{1}{2}$, this equation is shown to have a PDE scaling limit, from which distributional limit theorems for the RDE are derived. Applied to the series-parallel graph, the results imply that $N^{-1/3} \log R^{(N)}$ has a nondegenerate limit when $p = \frac{1}{2}$, as conjectured by Addario-Berry, Cairns, Devroye, Kerriou, and Mitchell [1]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11036 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph Morfe, Peter S. Probability Analysis of PDEs This paper analyzes a class of recursive distributional equations (RDE's) proposed by Gurel-Gurevich [17] and involving a bias parameter $p$, which includes the logarithm of the resistance of the series-parallel graph. A discrete-time evolution equation resembling a quasilinear Fisher-KPP equation is derived to describe the CDF's of solutions. When the bias parameter $p = \frac{1}{2}$, this equation is shown to have a PDE scaling limit, from which distributional limit theorems for the RDE are derived. Applied to the series-parallel graph, the results imply that $N^{-1/3} \log R^{(N)}$ has a nondegenerate limit when $p = \frac{1}{2}$, as conjectured by Addario-Berry, Cairns, Devroye, Kerriou, and Mitchell [1]. |
| title | Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2511.11036 |