Universality in the target arrival statistics of non-conservative search processes
Abstract
Stochastic search processes in which agents are continuously inserted into and removed from a target search domain are fundamental to a wide class of physical and artificial systems. The theory of such non-conservative search processes is, however, much less developed than for search processes with a fixed number of agents. Recent work on the non-conservative case has characterized the fastest first-passage time (FPT) to a target, after which the search process is terminated. Here we consider the complementary problem of continuous arrival statistics to a target. In particular, we derive the full time-dependent distribution of target arrivals, which we use to show that the inter-arrival time distribution is exactly exponential in steady-state regardless of the underlying search process. Thus, counterintuitively, the arrival statistics of non-conservative search are both universal and much simpler than sequential search-and-capture processes involving a fixed number of agents. Our methods also yield an exact expression for the fastest FPT distribution with time-dependent insertion, thereby generalizing previous work on fixed-rate insertion for non-conservative search. Altogether these results have major implications for target resource accumulation, wherein the delivery of resources is counter-balanced by their downstream consumption.
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BibTeX
@misc{giralbarajas2026universality,
title = {{Universality in the target arrival statistics of non-conservative search processes}},
author = {José Giral-Barajas and Samantha Linn and Paul C. Bressloff},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.16025},
eprint = {2606.16025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16025},
}
RIS
TY - UNPB
TI - Universality in the target arrival statistics of non-conservative search processes
AU - José Giral-Barajas
AU - Samantha Linn
AU - Paul C. Bressloff
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.16025
UR - https://arxiv.org/abs/2606.16025
AB - Stochastic search processes in which agents are continuously inserted into and removed from a target search domain are fundamental to a wide class of physical and artificial systems. The theory of such non-conservative search processes is, however, much less developed than for search processes with a fixed number of agents. Recent work on the non-conservative case has characterized the fastest first-passage time (FPT) to a target, after which the search process is terminated. Here we consider the complementary problem of continuous arrival statistics to a target. In particular, we derive the full time-dependent distribution of target arrivals, which we use to show that the inter-arrival time distribution is exactly exponential in steady-state regardless of the underlying search process. Thus, counterintuitively, the arrival statistics of non-conservative search are both universal and much simpler than sequential search-and-capture processes involving a fixed number of agents. Our methods also yield an exact expression for the fastest FPT distribution with time-dependent insertion, thereby generalizing previous work on fixed-rate insertion for non-conservative search. Altogether these results have major implications for target resource accumulation, wherein the delivery of resources is counter-balanced by their downstream consumption.
ER -