Scaling limits of interacting diffusions in domains

AbstractPitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.

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How far will silicon nanocrystals push the scaling limits of NVMs technologies?

IEEE International Electron Devices Meeting 2003 ◽

10.1109/iedm.2003.1269352 ◽

2004 ◽

Cited By ~ 39

Author(s):

B. De Salvo ◽

C. Gerardi ◽

S. Lombardo ◽

T. Baron ◽

L. Perniola ◽

...

Keyword(s):

Silicon Nanocrystals ◽

Scaling Limits

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Liouville first-passage percolation: Subsequential scaling limits at high temperature

The Annals of Probability ◽

10.1214/18-aop1267 ◽

2019 ◽

Vol 47 (2) ◽

pp. 690-742 ◽

Cited By ~ 10

Author(s):

Jian Ding ◽

Alexander Dunlap

Keyword(s):

High Temperature ◽

Scaling Limits ◽

First Passage Percolation ◽

First Passage

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CMOS SRAM scaling limits under optimum stability constraints

2013 IEEE International Symposium on Circuits and Systems (ISCAS2013) ◽

10.1109/iscas.2013.6572132 ◽

2013 ◽

Cited By ~ 7

Author(s):

Adam Makosiej ◽

Olivier Thomas ◽

Amara Amara ◽

Andrei Vladimirescu

Keyword(s):

Scaling Limits ◽

Optimum Stability

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On the effect of heterogeneity on flocking behavior and systemic risk

Statistics & Risk Modeling ◽

10.1515/strm-2016-0013 ◽

2017 ◽

Vol 34 (3-4) ◽

Cited By ~ 1

Author(s):

Fei Fang ◽

Yiwei Sun ◽

Konstantinos Spiliopoulos

Keyword(s):

Asymptotic Formula ◽

Case Studies ◽

Systemic Risk ◽

Mean Field ◽

Loss Probability ◽

Loss Distribution ◽

Interacting Diffusions ◽

The Mean ◽

Laplace Asymptotics ◽

Flocking Behavior

AbstractThe goal of this paper is to study organized flocking behavior and systemic risk in heterogeneous mean-field interacting diffusions. We illustrate in a number of case studies the effect of heterogeneity in the behavior of systemic risk in the system, i.e., the risk that several agents default simultaneously as a result of interconnections. We also investigate the effect of heterogeneity on the “flocking behavior” of different agents, i.e., when agents with different dynamics end up following very similar paths and follow closely the mean behavior of the system. Using Laplace asymptotics, we derive an asymptotic formula for the tail of the loss distribution as the number of agents grows to infinity. This characterizes the tail of the loss distribution and the effect of the heterogeneity of the network on the tail loss probability.

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