arXiv subjects

Daecheol Kim

Publications and source records attributed to Daecheol Kim.

4 recordsLinked to original sources

arXiv2026

Inhomogeneous Long-Range First-Passage Percolation in a Random Vertex Environment

We study inhomogeneous long-range first-passage percolation on $\mathbb{Z}^d$ with edge passage times $\lVert x-y\rVert^αω_{xy}/(V_xV_y)$, where vertex weights have polynomial upper-tail exponent $γ$ and edge noises have polynomial lower-tail exponent $θ$ at zero. The model interpolates between long-range first-passage percolation and scale-free percolation and exhibits competition between reusable heavy-vertex hubs and pair-specific small-noise bridges. We conjecture an eight-regime phase diagram, organized into five growth phases governed by $q_{\rm hub}=d/γ$ and $q_{\rm edge}=d/θ$. For $T_n=T(0,\lceil nx\rceil)$, we prove upper bounds of the conjectured order in every regime and matching lower bounds in phases I and II. Specifically, $T_n=0$ a.s. when $α<q_{\rm hub}\vee q_{\rm edge}$, while $T_n=Θ_{\mathbb{P}}(1)$ when $q_{\rm hub}\vee q_{\rm edge}<α<2q_{\rm hub}$. In the edge-dominated intermediate regime, $T_n=O_{\mathbb{P}}((\log n)^{Δ_{\rm III}+\varepsilon})$, where $Δ_{\rm III}=\log 2/\log(2q_{\rm edge}/α)$. The two power-law regimes satisfy $T_n=O_{\mathbb{P}}(n^{α-2q_{\rm hub}+\varepsilon})$ and $T_n=O_{\mathbb{P}}(n^{α-2q_{\rm edge}+\varepsilon})$, and the linear regime satisfies $T_n=O_{\mathbb{P}}(n)$. All upper bounds are constructive, based on hub-chain and binary edge-bridge multiscale constructions.

math.PR↗

arXiv2026

First-Passage Percolation on Spread-out line graphs: Microscopic Regime

We study first-passage percolation on the $\ell$-spread-out line graph, where each vertex $i\in\{0,1,\dots,n\}$ is connected to all others at distance at most $\ell$. Here, we focus on the microscopic regime, with $\ell$ fixed as $n\to\infty$. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time $T_n$ from $0$ to $n$. If the weight distribution has finite variance or a heavy tail with exponent above $2/\ell_c$ where $\ell_c=\ell(\ell+1)/2$, then $T_n$ satisfies a Gaussian CLT with $\sqrt{n}$ scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that $T_n$, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of $T_n$.

math.PR↗

arXiv2026

Phase Transition and Fluctuation Results for First-Passage Percolation on Spread-Out Cycle Graphs

We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $ω_e \sim \mathrm{Exp}(1)^{1/θ}$ to the edges for $θ>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \ll \ell \ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump--Mode--Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\ell\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\ell \approx λn$ for $λ\in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\ell \asymp n/\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.

math.PR↗

arXiv2021

Energy landscape of the two-component Curie-Weiss-Potts model with three spins

In this paper, we investigate the energy landscape of the two-component spin systems, known as the Curie-Weiss-Potts model, which is a generalization of the Curie-Weiss model consisting of $q\ge3$ spins. In the energy landscape of a multi-component model, the most important element is the relative strength between the inter-component interaction strength and the component-wise interaction strength. If the inter-component interaction is stronger than the component-wise interaction, we can expect all the components to be synchronized in the course of metastable transition. However, if the inter-component interaction is relatively weaker, then the components will be desynchronized in the course of metastable transition. For the two-component Curie-Weiss model, the phase transition from synchronization to desynchronization has been precisely characterized in studies owing to its mean-field nature. The purpose of this paper is to extend this result to the Curie-Weiss-Potts model with three spins. We observe that the nature of the phase transition for the three-spin case is entirely different from the two-spin case of the Curie-Weiss model, and the proof as well as the resulting phase diagram is fundamentally different and exceedingly complicated.

math.PR↗