arXiv subjects

Osama Abuzaid

Publications and source records attributed to Osama Abuzaid.

4 recordsLinked to original sources

arXiv2026

Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)

We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion (DBM) on the circle, for a fixed number of particles as the coupling parameter $β=8/κ$ tends to $+\infty$. Zero-energy systems correspond to the Calogero-Moser-Sutherland integrable system. We also characterize the large-time behavior of finite-energy systems: zero-energy systems approach exponentially fast a static equally-spaced configuration, while finite-energy systems may have polynomial convergence rates, and the system may never become static. We use our DBM result to derive a finite-time LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_κ$, as $κ$ tends to $0+$, with good rate function being the multiradial Loewner energy. Using a derivative estimate for the radial Loewner map in terms of the energy of its driving function, we show that finite-energy multiradial Loewner hulls are disjoint unions of simple curves, except at their common endpoint.

math.PR↗

arXiv2026

Large deviations of SLE(0+) variants in the capacity parameterization

We prove large deviation principles (LDPs) for full chordal, radial, and multichordal SLE(0+) curves parameterized by capacity. The rate function is given by the appropriate variant of the Loewner energy. There are two key novelties in the present work. First, we strengthen the topology in the known chordal LDPs into the topology of full parameterized curves including all curve endpoints. We also obtain LDPs in the space of unparameterized curves. Second, we address the radial case, which requires in part different methods from the chordal case, due to the different topological setup. We establish our main results via proving an exponential tightness property and combining it with detailed curve escape probability estimates, in the spirit of exponentially good approximations in LDP theory. In the radial case, additional work is required to refine the estimates appearing in the literature. Notably, since we manage to prove a finite-time LDP in a better topology than in earlier literature, escape energy estimates follow as a consequence of the escape probability estimates.

math.PR↗

arXiv2026

Multiradial Schramm-Loewner evolution: Infinite-time large deviations and transience

In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(κ)$, as $κ\to 0$, with good rate function being the multiradial Loewner energy. Here, we extend this result to infinite time in the topology of common-capacity-parameterized curves, and streamline the proof. A main step is to derive detailed escape probability estimates for multiradial SLE$(κ)$ curves in the common parameterization, which extend the single-curve estimates achieved in [AP26]. As a by-product, we also get that multiradial SLE$(κ)$ curves, with $κ\leq 8/3$, are transient at their common terminal point, generalizing [FL15, HL21]. As a corollary to the LDP result, we obtain explicit asymptotics of the Brownian loop measure interaction term for finite-energy radial multichords, which is linear in the capacity-time and coincides with a certain choice of a cocycle for the Virasoro algebra.

math.PR↗

arXiv2025

Precompactness of sequences of random variables and random curves revisited

This paper studies when a sequence of probability measures on a metric space admit subsequential weak limits. A sufficient condition called sequential tightness is formulated, which relaxes some assumptions for asymptotic tightness used in the Prokhorov -- Le Cam theorem. The proof only uses elementary tools from probability theory. Sequential tightness gives means to characterize the precompact collections of random curves on a compact geodesic metric space in terms of an annulus crossing condition, which generalizes the one by Aizenman and Burchard by allowing estimates for annulus crossing probabilities to be non-uniform over the modulus of annuli.

math.PR↗