arXiv subjects

Lucas Rey

Publications and source records attributed to Lucas Rey.

7 recordsLinked to original sources

arXiv2026

Massive holomorphicity of near-critical dimers and sine-Gordon model

In this paper, we consider the near-critical dimer model in the setup of isoradial superpositions with Temperleyan boundary conditions. We show that the centered height function converges as the mesh size tends to zero to a limiting field which agrees with the (electromagnetically tilted) sine-Gordon model, whose derivative correlations are described by Grassmann variables (or equivalently determinants involving a massive Dirac operator). This answers a longstanding question in the field. A crucial part of the work is to develop a notion of discrete massive holomorphic functions and the tools to study such functions, in particular finding an exact discrete form of the massive Cauchy--Riemann equations, which is satisfied by the inverse Kasteleyn matrix. In comparison with previous studies, a key novelty of this part of our work is that the mass is not only allowed to be non-constant but can be complex-valued.

math.PR↗

arXiv2026

Classification of (non)-frustrated 2D Ising models in genus 1 on isoradial graphs

We prove a complete classification of 2D Ising models defined on isoradial graphs, frustrated or not, whose underlying spectral curve has genus 1. As a specific case, we recover Baxter's Z-invariant Ising model, thus extending his class of models to real coupling constants. We identify two additional families of models, both having non-Harnack spectral curves. We show that in all cases the spectral curve is maximal. Moreover, each family undergoes an algebraic phase transition, in the sense that the genus changes from one to zero, explaining the different behaviors observed in the physics literature. In our proof, we use properties of the spectral curve and Fock's approach. This yields a natural framework for a further systematic study of the frustrated Ising model, in particular for proving local formulas. In the course of the proof, we also identify Fock's dimer models corresponding to real algebraic curves of genus 1, and to real dimer models. As an example of application of our main result, we prove a full classification of the frustrated Ising model on the triangular lattice.

math-ph↗

arXiv2025

A Performance Analysis of You Only Look Once Models for Deployment on Constrained Computational Edge Devices in Drone Applications

Advancements in embedded systems and Artificial Intelligence (AI) have enhanced the capabilities of Unmanned Aircraft Vehicles (UAVs) in computer vision. However, the integration of AI techniques o-nboard drones is constrained by their processing capabilities. In this sense, this study evaluates the deployment of object detection models (YOLOv8n and YOLOv8s) on both resource-constrained edge devices and cloud environments. The objective is to carry out a comparative performance analysis using a representative real-time UAV image processing pipeline. Specifically, the NVIDIA Jetson Orin Nano, Orin NX, and Raspberry Pi 5 (RPI5) devices have been tested to measure their detection accuracy, inference speed, and energy consumption, and the effects of post-training quantization (PTQ). The results show that YOLOv8n surpasses YOLOv8s in its inference speed, achieving 52 FPS on the Jetson Orin NX and 65 fps with INT8 quantization. Conversely, the RPI5 failed to satisfy the real-time processing needs in spite of its suitability for low-energy consumption applications. An analysis of both the cloud-based and edge-based end-to-end processing times showed that increased communication latencies hindered real-time applications, revealing trade-offs between edge (low latency) and cloud processing (quick processing). Overall, these findings contribute to providing recommendations and optimization strategies for the deployment of AI models on UAVs.

cs.DC↗

arXiv2024

On loops in the complement to dimers

We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e. perfect matchings) on the hexagonal lattice. The complement to a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop $O(1)$ model at $x=\infty$. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path. Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in the dimer theory imply that such hexagons appear with a positive density. Up to some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton--Keane argument, to exclude existence of more than one bi-infinite path.

math.PR↗

arXiv2024

Conformally invariant boundary arcs in double dimers

We consider two different versions of the double dimer model on a planar domain, where we either fold a single dimer cover on a symmetric domain onto itself across the line of symmetry, or we superimpose two independent dimer covers on two, almost identical, domains that differ only on a certain portion of the boundary. This results in a collection of loops and doubled edges that, unlike in the classical double dimer case of Kenyon, are accompanied by arcs emanating from the line of symmetry or the chosen portion of the boundary. We argue that these arcs together with the associated height function satisfy a discrete version of the coupling of Qian and Werner between the Arc loop ensemble (ALE) and two different variants of the Gaussian free field (with Dirichlet and Neumann boundary conditions). We also show that certain statistics of the arcs (when the loops are disregarded from the picture) converge to conformally invariant quantities in the small-mesh scaling limit, and moreover the limits are the same for the two versions of the model, and equal to the corresponding statistics of the arc loop ensemble (ALE). This gives evidence to the conjecture of [7] (that concerns one of these models).

math.PR↗

arXiv2024

Can one condition a killed random walk to survive?

We consider the simple random walk on $\mathbb{Z}^d$ killed with probability $p(|x|)$ at site $x$ for a function $p$ decaying at infinity. Due to recurrence in dimension $d=2$, the killed random walk (KRW) dies almost surely if $p$ is positive, while in dimension $d \geq 3$ it is known that the KRW dies almost surely if and only if $\int_0^{\infty}rp(r)dr = \infty$, under mild technical assumptions on $p$. In this paper we consider, for any $d \geq 2$, functions $p$ for which the KRW dies almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion $(\Lambda_R)_{R \in \mathbb{N}}$ of $\mathbb{Z}^d$, does the KRW conditioned to leave $\Lambda_R$ before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for $p(r) = o(r^{-2})$, and that it is not for $p(r) = \min(1, r^{-\alpha})$ for $\alpha \in (14/9,2)$. This question is connected to branching random walks and the infinite snake. More precisely, in dimension $d=4$, the infinite snake is related to the KRW with $p(r) \asymp (r^2\log(r))^{-1}$, therefore our results imply that the infinite snake conditioned to avoid the origin in four dimensions is well-defined.

math.PR↗

arXiv2024

The Doob transform and the tree behind the forest, with application to near-critical dimers

The Doob transform technique enables the study of a killed random walk (KRW) via a random walk (RW) with transition probabilities tilted by a discrete massive harmonic function. The main contribution of this paper is to transfer this powerful technique to statistical mechanics by relating two models, namely random rooted spanning forests (RSF) and random spanning trees (RST), and provide applications. More precisely, our first main theorem explicitly relates models on the level of partition functions, and probability measures, in the case of finite and infinite graphs. Then, in the planar case, we also rely on the dimer model: we introduce a killed and a drifted dimer model, extending to this general framework the models introduced in [Chh12,dT20]. Using Temperley's bijection between RST and dimers, this allows us to relate RSF to dimers and thus extend partially this bijection to RSF. As immediate applications, we give a short and transparent proof of Kenyon's result stating that the spectral curve of RSF is a Harnack curve, and provide a general setting to relate discrete massive holomorphic and harmonic functions. The other important application consists in proving universality of the convergence of the near-critical loop-erased RW, RST and dimer models by extending the results of [Chh12,CW21,HSB22] from the square lattice to any isoradial graphs: we introduce a loop erased RW, RST and dimer model on isoradial discretizations of any simply connected domain and prove convergence in the massive scaling limit towards continuous objects described by a massive version of SLE2.

math.PR↗