Bayesian Monitoring of a Diffusive Particle in One Dimension
Abstract
We study the Bayesian monitoring of a single particle diffusing along a line, while an observer tracks it continuously using noisy measurements at each spatial point. The average Shannon entropy S(t)̅ associated with the posterior distribution of the particle's position quantifies the uncertainty that remains after conditioning on the sequence of measurements. We map the problem to the calculation of the moments of the partition functions of directed polymers in 1 + 1 dimensions. For a constant monitoring rate, the entropy remains bounded for any finite measurement intensity and can be quantified using exact results from the Kardar-Parisi-Zhang equation, providing both the saturation value and the asymptotic behavior for long times. For a monitoring intensity that decreases according to a power law ∼ t−α, two distinct asymptotic regimes emerge: S(t)̅ ≃ 1/2 ln t for α> 1/2 and S(t)̅ ≃ αln t for 0 < α< 1/2. Both results are obtained using the replica method, which maps the problem onto an attractive Lieb-Liniger Hamiltonian with a time-dependent coupling: the two regimes can be understood from a perturbative expansion around free diffusion and around the attractive Lieb-Liniger ground state, respectively. We discuss the limiting case α= 1/2 and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.
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BibTeX
@misc{gerbino2026bayesian,
title = {{Bayesian Monitoring of a Diffusive Particle in One Dimension}},
author = {Federico Gerbino and Guido Giachetti and Pierre Le Doussal and Andrea De Luca},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28655},
eprint = {2609.28655},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05844393v1},
}
RIS
TY - UNPB
TI - Bayesian Monitoring of a Diffusive Particle in One Dimension
AU - Federico Gerbino
AU - Guido Giachetti
AU - Pierre Le Doussal
AU - Andrea De Luca
PY - 2026
DA - 2026/09/23
PB - HAL
DO - 10.48550/arxiv.2609.28655
UR - https://hal.science/hal-05844393v1
AB - We study the Bayesian monitoring of a single particle diffusing along a line, while an observer tracks it continuously using noisy measurements at each spatial point. The average Shannon entropy S(t)̅ associated with the posterior distribution of the particle's position quantifies the uncertainty that remains after conditioning on the sequence of measurements. We map the problem to the calculation of the moments of the partition functions of directed polymers in 1 + 1 dimensions. For a constant monitoring rate, the entropy remains bounded for any finite measurement intensity and can be quantified using exact results from the Kardar-Parisi-Zhang equation, providing both the saturation value and the asymptotic behavior for long times. For a monitoring intensity that decreases according to a power law ∼ t^−α, two distinct asymptotic regimes emerge: S(t)̅ ≃ 1/2 ln t for α> 1/2 and S(t)̅ ≃ αln t for 0 < α< 1/2. Both results are obtained using the replica method, which maps the problem onto an attractive Lieb-Liniger Hamiltonian with a time-dependent coupling: the two regimes can be understood from a perturbative expansion around free diffusion and around the attractive Lieb-Liniger ground state, respectively. We discuss the limiting case α= 1/2 and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.
ER -