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Abstract In this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well‐known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi‐type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, and so forth. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein–Rutman eigenvalue and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if .
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This content will become publicly available on March 1, 2026
Propagation of Chaos for Point Processes Induced by Particle Systems with Mean-Field Drift Interaction
Abstract We study the asymptotics of the point process induced by an interacting particle system with mean-field drift interaction. Under suitable assumptions, we establish propagation of chaos for this point process: It has the same weak limit as the point process induced by i.i.d. copies of the solution of a limiting McKean–Vlasov equation. This weak limit is a Poisson point process whose intensity measure is related to classical extreme value distributions. In particular, this yields the limiting distribution of the normalized upper order statistics.
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- Award ID(s):
- 2206062
- PAR ID:
- 10617311
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Journal of Theoretical Probability
- Volume:
- 38
- Issue:
- 1
- ISSN:
- 0894-9840
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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