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Imagine a configuration of particles that is neither crystal-like nor completely liquid- or gas-like: it behaves
at short length scales like a gas and at large length scales more like a crystal; such a configuration is a
typical example of hyperuniform organization 1 . All perfect crystals, perfect quasicrystals and special
disordered systems are hyperuniform. Over the last decade, such forms of many-body organization have
attracted considerable attention due to their discovery in both classical and quantum systems, both
equilibrium and nonequilibrium, with examples as diverse as the distribution of prime numbers or the
optical cells in avian photoreceptors. Apart from their natural occurrence, hyperuniform states have also
become important in designing meta-materials with optimal photonic, phononic, and transport properties,
particularly displaying the isotropic photonic band-gap in disordered systems. Despite this ubiquity,
hyperuniformity has rarely been studied in non-equilibrium stochastic systems with non-conserved particle
number — precisely the regime relevant for biological and ecological pattern formation. In this talk, after
introducing hyperuniformity in general, I shall discuss our recent work on the occurrence of effective
hyperuniformity in stochastic Turing patterns in reaction-diffusion systems. Stochastic reaction-diffusion
systems provide a generic framework for modeling biological and ecological processes driven by
demographic noise, successfully explaining pattern formation in several natural processes without
requiring the deterministic Turing instability criterion. A key finding is that while instantaneous
configurations are not hyperuniform, time-integrated population densities exhibit effective hyperuniformity.
Using the linear noise approximation, we analytically predict this emergent large-scale spatial order, and
show that the degree of hyperuniformity is tunable across orders of magnitude, establishing that
demographic noise, without any fine-tuning of reaction parameters, is sufficient to generate macroscopic
spatial organization in a broad class of non-equilibrium biological systems. |