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Arnab Sen is a mathematician working in probability theory and mathematical physics. His own biography identifies a Berkeley doctorate completed in 2010, subsequent research at Cambridge, and work at the University of Minnesota. His research includes random graphs, random matrices and disordered models in statistical physics.^[ref]

With Nathanaël Berestycki and Christophe Garban, he developed a new state space and topology for the coalescing Brownian flow and proved an probabilistic invariance principle for coalescing random walks under a finite-variance assumption. The theorem connects discrete particle systems with a continuum stochastic flow.^[ref]

The physical picture is simple even though convergence of an entire flow is subtle: particles wander separately until meeting, then continue as one. Tracking only the position of one particle misses when two histories merge. A framework for the whole collection must retain that interaction.

Christophe Garban is a mathematician whose research connects probability theory with statistical physics. His institutional and personal pages identify work on percolation, phase transitions and scaling limits, with a professorship at Lyon 1 beginning in 2014.^[ref]

His work with collaborators constructs limiting measures for pivotal sites, clusters and interfaces in critical planar percolation. A pivotal site is a location whose state can change a macroscopic connectivity event. Counting such locations requires a scale-dependent normalization before a nontrivial continuum limit emerges. The resulting measures make sensitivity to microscopic changes part of the limiting geometry.^[ref]

With Nathanaël Berestycki and Arnab Sen, he also constructed a state-space description yielding convergence from coalescing random walks to a coalescing Brownian flow with only finite variance of the increments. This connects his study of continuum random geometry with interacting stochastic paths.^[ref]

Portrait: Institut universitaire de France, identified portrait and research profile.

A coalescing Brownian flow describes particles that follow independent one-dimensional Brownian motions until they meet, after which they move together. For any finite collection of deterministic space-time starting points, this rule specifies a consistent interacting system. Extending it to trajectories from all space-time points requires a suitable state space: separate one-particle distributions do not record which histories merge.^[ref]

Two particles give an illuminating calculation. Start them at $0$ and $a>0$, with the normalization $\mathrm{Var}(B_t)=t$, and let $T$ be their first meeting time. Before meeting, their separation has the first-hit law of $a+\sqrt{2}{W}_t$, because the difference of independent Brownian increments has twice their variance. The Brownian reflection principle giveswhere $\Phi$ is the standard normal distribution function. For $a=1$ and $t=1$, the meeting probability is about $0.4795$. After a meeting the two positions are identical, although each position separately still has the appropriate Brownian marginal law. Continuing to drive both particles independently would describe a different model.

The flow connects microscopic merging trajectories with macroscopic random geometry. Nathanaël Berestycki, Christophe Garban and Arnab Sen developed a state-space description in which rescaled coalescing random walks converge under a finite-variance assumption. Their result improves the moment requirements needed by earlier descriptions; it does not identify the origin of the subject with that later theorem.^[ref]

Nathanaël Berestycki works in probability theory, connecting stochastic motion with random geometry. His University of Vienna homepage identifies research on Brownian motion, the Gaussian free field, Liouville quantum gravity and interacting particle systems.^[ref]

With Christophe Garban and Arnab Sen, he introduced a topology for the coalescing Brownian flow and obtained convergence from coalescing random walks under finite variance of the increments. The state-space choice makes the continuum limit accessible without stronger moment assumptions previously needed in other descriptions.^[ref]

A useful distinction is between convergence of finitely many trajectories and convergence of a whole random flow. Starting particles at many locations creates a shared pattern of meetings and mergers. A continuum description has to preserve the consistency of those histories, rather than merely fit the distribution of one endpoint.