Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.11589

On the time constant of high dimensional first passage percolation, revisited

Antonio Auffinger·Si Tang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In [2], it was claimed that the time constant $μ_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $μ_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(τ_{e})_{e\in \mathbb E^{d}}$ are i.i.d., with a common c.d.f. $F$ satisfying $\left|\frac{F(x)}{x}-a\right| \le \frac{C}{|\log x|}$ for some constants $a, C$ and sufficiently small $x$. However, the proof of the upper bound, namely, Equation (2.1) in [2] \begin{align} \limsup_{d\to\infty} \frac{μ_{d}(e_{1})ad}{\log d} \le \frac{1}{2} \end{align} is incorrect. In this article, we provide a different approach that establishes this inequality. As a side product of this new method, we also show that the variance of the non-backtracking passage time to the first hyperplane is of order $o\big((\log d/d)^{2}\big)$ as $d\to \infty$ in the case of the when the edge weights are exponentially distributed.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Antonio Auffinger, Si Tang. 2025-01-20. On the time constant of high dimensional first passage percolation, revisited. https://arxiv.org/abs/2501.11589

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Local well-posedness of general mean field game master equations

This paper presents a generic approach for establishing mean field game master equations, applicable whenever the mean field equilibrium can be characterized by a McKean-Vlasov forward-backward stochastic differential equation system. The core of our approach is a representation formula for the first-order Lions derivative of the decoupling field of this forward-backward SDE system. We then employ a bootstrap argument to recursively compute its higher-order derivatives. To demonstrate the method's versatility, we establish the local well-posedness for master equations in three distinct models: extended mean field games, mean field games with volatility control, and mean field games with a major player.

math.PR↗
arXiv2026

Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes

This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)β(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $β$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv δ_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

math.PR↗
arXiv2026

Small-time annealed large deviations principle for one-dimensional diffusions in a random environment

In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-ρ(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{ρ(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $ρ$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The Itô--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.

math.PR↗