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arXiv:2609.28655

Bayesian Monitoring of a Diffusive Particle in One Dimension

Federico Gerbino, Guido Giachetti, Pierre Le Doussal, Andrea De Luca

Published Sep 23, 2026• 0 citations
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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 $\overline{S(t)}$ 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 $\sim t^{-α}$, two distinct asymptotic regimes emerge: $\overline{S(t)} \simeq \tfrac12 \ln t$ for $α> 1/2$ and $\overline{S(t)} \simeq α\ln t$ 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.

DOI
10.48550/arxiv.2609.28655

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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)‾\overline{S(t)}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+11 + 11+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\sim t^{-α}∼t−α, two distinct asymptotic regimes emerge: S(t)‾≃12ln⁡t\overline{S(t)} \simeq \tfrac12 \ln tS(t)​≃21​lnt for α>1/2α> 1/2α>1/2 and S(t)‾≃αln⁡t\overline{S(t)} \simeq α\ln tS(t)​≃αlnt for 0<α<1/20 < α< 1/20<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=1/2α= 1/2α=1/2 and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.
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Abstract We revisit the Lieb–Liniger model for n bosons in one dimension with attractive delta interaction in a half-space with diagonal boundary conditions. This model is integrable for the arbitrary value of , the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case , with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar–Parisi–Zhang equation for the growth of a one-dimensional interface of height , on the half-space with boundary condition and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b , for the integer exponential (one-point) moments of the KPZ height field . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for where the PDF is given by the GSE Tracy–Widom distribution (ii) bound for , where the PDF is a Gaussian. At the critical point , the PDF is given by the GOE Tracy–Widom distribution.
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