EXPLORING RANDOMNESS
Doubly stochastic continuous time random walk
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Since its introduction some 60 years ago, the Montroll-Weiss continuous time random walk has found numerous applications due its ease of use and ability to describe both regular and anomalous diffusion. Yet, despite its broad applicability and generality, the model cannot account for effects coming from random diffusivity fluctuations, which have been observed in the motion of asset prices and molecules. To bridge this gap, we introduced a doubly stochastic version of the model in which waiting times between jumps are replaced with a fluctuating jump rate [Phys. Rev. Research 6, L012033, 2024].
We show that this newly added layer of randomness gives rise to a rich phenomenology while keeping the model fully tractable, allowing us to explore general properties and illustrate them with examples. In particular, we show that the model provides an alternative pathway to Brownian yet non-Gaussian diffusion, which has been observed and explained via diffusing diffusivity approaches.
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Loss of percolation transition in the presence of simple tracer-media interactions
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Random motion in disordered media is sensitive to the presence of obstacles which prevent atoms, molecules, and other particles from moving freely in space. When obstacles are static, a transition between confined motion and free diffusion occurs at a critical obstacle density: the percolation threshold. To test if this conventional wisdom continues to hold in the presence of simple tracer-media interactions from the type seen in recent experiments, we introduced the Sokoban random walk [Phys. Rev. Research 5, L042015, 2023]. Akin to the protagonist of an eponymous video game, the Sokoban has an ability to push single obstacles that block its path. While one expects this will allow the Sokoban to venture further away, we surprisingly find that this is not always the case. Indeed, as it moves on a 2d lattice—pushing obstacles around—the Sokoban always confines itself after traveling a characteristic distance that is set by the initial density of obstacles. Consequently, the percolation transition is lost. This finding breaks from the ruling ant in a labyrinth paradigm, vividly illustrating that even weak and localized tracer-media interactions cannot be neglected when coming to understand transport phenomena.
Universal statistics for Min-Max/Max-Min of large random matrices
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The Max-Min and Min-Max of matrices arise prevalently in science and engineering. However, in many real-world situations, the computation of the Max-Min and Min-Max is challenging as matrices are large and full information about their entries is lacking. In this duo of papers [Phys. Rev. E 100, 020104(R), 2019; Phys. Rev. E 100, 022129, 2019], we took a statistical-physics approach and established limit laws—akin to the central limit theorem—for the Max-Min and Min-Max of large random matrices. The limit laws assert that Gumbel statistics emerge irrespective of the matrix entries' distribution. The universality and simplicity of this result makes us hopeful that it will find various applications.
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Occupancy correlations in the asymmetric simple inclusion process
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The asymmetric simple inclusion process (ASIP) is a lattice-gas model for unidirectional transport with irreversible aggregation [Phys. Rev. Lett. 109, 020603, 2012; Phys. Rev. E 84, 041101, 2011]. To date, the analytical tractability of the model has been rather limited: while the average particle density is easy to compute, very little is known about the (full) joint occupancy distribution of particles. To partially bridge this gap, we studied occupancy correlations in the ASIP [Phys. Rev. E 100, 042109, 2019]. An exact formula for the covariance matrix of the occupancy vector at the steady-state was derived and corroborated against numerical simulations.
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