Abstract: Proximity measurements probe whether pairs of particles are close to one another, regardless of the absolute location of the pair in space. We consider the consequences of making a random sequence of proximity measurements on an initially fluid system composed of many distinguishable quantum particles, focusing on the case of full post-selection. Via a self-consistent mean-field theory, we find that such measurements lead to an unusual kind of particle organization: with a prevalence that increases over time, particles mutually localize one another in space, thus constituting an ever-thickening macroscopic, correlated assembly. Owing to its structural randomness, however, when viewed macroscopically this assembly retains the homogeneity of the original fluid. Ultimately, all particles localize and the distribution of their localization lengths stabilizes with a characteristic lengthscale determined by the measurement rate. The stable distribution of these lengths is governed by a universal scaling form that has arisen in several distinct settings.
Work done with Pushkar Mohile (arXiv:2507.23085)
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