关于几个特殊的有向最后一段渗流模型

Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation

Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics, however the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on $\mathbb{Z}^2$ with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integr…

关于首程渗流时间常数的一个下界

On a Lower Bound for the Time Constant of First-Passage Percolation

本文考虑了$\mathbb Z^d (d\ge 2)$上的Bernoulli首通渗流问题。也就是说，边缘通过时间在概率为$1-p$的情况下独立地取为1，否则取为0。设${μ(p)}$为时间常数。本文利用Russo公式证明了对所有$ 0\leq p_1<p_2< 1$，都有\[ μ(p_1)-μ({p_2})\ge \frac{μ(p_2)}{1-p_2}(p_2-p_1)\].

The two-time distribution in geometric last-passage percolation

We study the two-time distribution in directed last passage percolation with geometric weights in the first quadrant. We compute the scaling limit and show that it is given by a contour integral of a Fredholm determinant.

首段渗流标度指数之间的普适关系

The universal relation between scaling exponents in first-passage percolation

在许多物理论文中已经证明，在整数格点上的普通首次通过渗流中，涨落指数$χ$和游荡指数$ξ$通过普适关系$χ=2ξ-1$联系在一起，而与维数无关。这有时被称为两个指数之间的KPZ关系。本文在假设指数在一定意义下存在的前提下，给出了该猜想的严格证明。

First passage percolation in the mean field limit

The Poisson clumping heuristic has lead Aldous to conjecture the value of the first passage percolation on the hypercube in the limit of large dimensions. Aldous' conjecture has been rigorously confirmed by Fill and Pemantle [Annals of Applied Prob- ability 3 (1993)] by means of a variance reduction trick. We present here a streamlined and, we believe, more natural proof based on ideas emerged in the study of Derrida's random energy models.

Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle

In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^α\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-α)/2}$, $0<α<1/3$.

具有几何随机变量的有向最后通过点渗流的最优尾估计

Optimal tail estimates for directed last passage site percolation with geometric random variables

本文得到了Johansson在[12]中所考虑的最后一个通道点渗流模型的最长上/右路径的适当标度长度的概率分布的最优一致下尾估计.估计值用于证明模型的下尾中等偏差结果。估计也意味着收敛的时刻，也提供了一个验证的普遍尺度律有关的纵向和横向波动的模型。

三角格子上临界首次通过渗流的Law of large numbers

Law of large numbers for critical first-passage percolation on the triangular lattice

我们研究了三角形格子$\mathbb{T}$上的（独立）第一次通过逾渗问题。用$t(v)$表示$\mathbb{T}$中位点$v$的通过时间，并假设$P(t(v)=0)=P(t(v)=1)=1/2$。$a_{0,n}$表示从$\textbf{0}$到$(n,0)$的通过时间，$b_{0,n}$表示从$\textbf{0}$到半平面$\{(x,y):x\geq n\}$的通过时间。证明了存在一个常数$0<μ<\infty$使得$n\rightarrow\infty$，$a_{0,n}/\log n\rightarrow μ$和$b_{0,n}/\log n\rightarrow μ/2$几乎必然.这一结果证实了Kesten和Zhang（Probab.理论关系Fields \textbf{107}：137- 160，1997）。该证明依赖于由Camia和Newman建立的$\mathbb{T}$上临界点渗流的全标度极限的存在。

非均匀首过渗流

Inhomogeneous first-passage percolation

我们研究了第一次通过渗流，其中在左半平面和右半平面中的边缘根据不同的分布被赋值。我们表明，所产生的非齐次的第一次通过过程的渐近增长服从形状定理，我们表示的限制形状的限制形状的齐次过程的两个权重分布。我们进一步表明，存在对分布的增长率在垂直方向上是严格大于增长率的均匀过程与两个分布中的任何一个，这对应于创建一个缺陷沿垂直轴的形式的“”。

末道渗流模型中的Busemann函数和平衡测度

Busemann functions and Equilibrium measures in last passage percolation models

二维渗流增长模型和一维粒子过程之间的相互作用已经成为有趣的数学现象的富有成效的来源。在本文中，我们开发的Hammersley最后一次通过渗流模型与独立同分布的Busemann函数的构造之间的连接。随机权，以及相关的（多类）相互作用流体系统的平衡（或时不变）测度的存在性、遍历性和唯一性.正如我们将看到的，在经典的哈默斯利模型中，每个点的权重为1，这种方法带来了一个新的，而最长的增加子序列问题的几何解决方案，以及一个中心极限定理的Busemann函数。

首次通过逾渗的两个定理的正温度版本

Positive temperature versions of two theorems on first-passage percolation

由于Talagrand（一个尾部的限制）和Benjamini-Kalai-Schramm（一个次线性方差限制）的第一次通过渗流的波动的估计被转录到正温度设置的随机Schroedinger算子。

单点上大偏差事件条件下的指数定向最后通过渗流的条件平坦指数模型

Conditional Flat Exponential Directed Last Passage Percolation under a One-Point Upper Large Deviation Event

该论文研究指数定向最后通过渗流在单点上大偏差事件下的条件模型，提出条件极限形状的预测并证明波动结果，发现对角线附近波动由共享高斯分量的布朗桥描述。

具有次线性距离方差的首次通过渗流

First Passage Percolation Has Sublinear Distance Variance

令$0<a<b<\infty$，对于$Z^d$的每个边$e$，令$ω_e=a$或$ω_e=b$，每个独立地具有概率1/2。这在$Z^d$的顶点上引起随机度量$\dist_ω$，称为第一次通过渗流。证明了对于$d>1$，从原点到顶点$v$，$|v|>2$的距离$dist_ω(0,v)$的方差以$C |v|/\log|v|$为界，其中$C=C(a,b,d)$是一个只依赖于$a$，$b$和$d$的常数.还讨论了一些相关的变体

i.i.d.中的非多边形极限形状示例。空间增长模型中的首次通过渗流和无限共存

Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

我们构造了i.i.d.的边权重分布.讨论了极限形状不是多边形且极值点在边界上任意稠密的$\mathbb{Z}^2$上的首过渗流问题。因此，相关的Richardson型增长模型可以支持可数无穷多个不同物种的共存，并且感染图具有无穷多个端点。

Time-time covariance for last passage percolation in half-space

This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance, then the first order correction to the covariance for the point-to-point model is the same as the one of the stationary model. In order to obtain the result, we first derive comparison inequalities of the last passage increments for different models. This is used to prove tightness of the point-to-point process as well as localization of the geodesics. Unlike for the full-space case, for half-space we…

Time Correlation Exponents in Last Passage Percolation

For directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices, let $T_{n}$ denote the last passage time from $(0,0)$ to $(n,n)$. We consider asymptotic two point correlation functions of the sequence $T_{n}$. In particular we consider ${\rm Corr}(T_{n}, T_{r})$ for $r\le n$ where $r,n\to \infty$ with $r\ll n$ or $n-r \ll n$. We show that in the former case ${\rm Corr}(T_{n}, T_{r})=Θ((\frac{r}{n})^{1/3})$ whereas in the latter case $1-{\rm Corr}(T_{n}, T_{r})=Θ((\frac{n-r}{n})^{2/3})$. The argument revolves around finer understanding of polymer geometry…

Regularity of the time constant for last passage percolation on complete directed acyclic graphs

We study the time constant $C(ν)$ of last passage percolation on the complete directed acyclic graph on the set of non-negative integers, where edges have i.i.d. weights with distribution $ν$ with support included in $\{-\infty\}\cup\mathbb{R}$. We show that $ν\mapsto C(ν)$ is strictly increasing in $ν$. We also prove that $C(ν)$ is continuous in $ν$ for a large set of measures $ν$. Furthermore, when $ν$ is purely atomic, we show that $C(ν)$ is analytic with respect to the weights of the atoms. In the special case of two positive atoms, it is an explicit rational function of these weights.

Stationary half-space last passage percolation

In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. We finally sh…

上尾条件下指数型最后通行渗流的单点涨落

One-point fluctuations for exponential last passage percolation under upper-tail conditioning

该研究针对上尾条件下的指数型定向最后通行渗流，确定了特定空间区域内及边界的单点涨落规律，扩展了相关工作，并通过串联队列对应关系揭示了非典型离站对网络其他位置的影响。

Random coalescing geodesics in first-passage percolation

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics'. Random coalescing geodesics have a range of nice asymptotic properties, such as asymptotic directions and linear Busemann functions. We show that random coalescing geodesics are (in some sense) dense in the space of geodesics. This allows us to extrapolate properties from random coalescing geodesi…

Tightness of Liouville first passage percolation for $γ\in (0,2)$

We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_t$ in the bulk, and related star-scale invariant metrics. For $γ\in (0,2)$ and $ξ= \fracγ{d_γ}$, where $d_γ$ is the Liouville quantum gravity dimension defined in [Ding-Gwynne18], we show that renormalized metrics $(λ_t^{-1} e^{ξp_t * h} ds)_{t \in (0,1)}$ are tight with respect to the uniform topology. In particular, we show that subsequential limits are bi-Hölder with respect to the Euclidean topology, obtain tail estimates for side-to-side distances an…

Geodesics Toward Corners in First Passage Percolation

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg and Hoffman (preprint, 2016), and others. However the main results do not cover geodesics in the direction of corners of the limit shape $\mathcal{B}$, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis d…

Francis Comets 的 Gumbel 末达渗流

Francis Comets' Gumbel last passage percolation

本文记录了 Francis Comets 关于 Gumbel 边权重的有向末达渗流与 log-gamma 有向聚合物模型联系的观察及相关渐近行为，以此纪念他对该领域的巨大贡献。

The distance exponent for Liouville first passage percolation is positive

Discrete Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{ξh(z)}$, where $h$ is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all $ξ> 0$. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size $2^n$ is typically at least $2^{αn}$ for an exponent $α> 0$ depending on $ξ$. This is a crucial input in the proof that LFPP admits non-trivial subsequential sca…

末次通过渗流模型的随机控制

Stochastic domination for the last passage percolation model

考虑了$\mathbb{Z}_+^{2}$上的竞争模型，该模型受有向末次通过渗流控制.提出了最后一段渗流的子树间的随机控制论。

On the combinatorics of last passage percolation in a quarter square and $\mathrm{GOE}^2$ fluctuations

In this note we give a(nother) combinatorial proof of an old result of Baik--Rains: that for appropriately considered independent geometric weights, the generating series for last passage percolation polymers in a $2n \times n \times n$ quarter square (point-to-half-line-reflected geometry) splits as the product of two simpler generating series---that for last passage percolation polymers in a point-to-line geometry and that for last passage percolation in a point-to-point-reflected (half-space) geometry, the latter both in an $n \times n \times n$ triangle. As a corollary, for iid geometric r…

入侵渗流与首达渗流的耦合

Coupling Invasion and First Passage Percolation

该研究通过构建入侵渗流与对数均匀通过时间首达渗流的耦合，为整数格伯努利键渗流的连续相变提供了新的等价条件，关联了原相变与顶点入侵概率消失的已知等价性。

非独立首过渗流模型中的时间常数

On the Time Constant in a Dependent First Passage Percolation Model

本文研究了Fontes和Newman提出的随机着色首通渗流模型。我们证明了该首段渗流模型的渐近形状连续依赖于着色律。证明使用了几个耦合，特别是贪婪的格子动物。

The height problem in first passage percolation

We consider the first passage percolation model in Z2 with a distribution F for 0 < F (0) < pc. In this paper, we solve the height problem.

第一通道渗流和竞争模型

First passage percolation and competition models

本文综述了首次通过渗流的各种结果和技术，首次通过渗流是一个在无限图上模拟扩散流体的随机过程。本文的后半部分着重于第一通道渗流与一类随机增长和竞争模型之间的联系。

Percolation and first-passage percolation on oriented graphs

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

非齐次第一行的半空间最后通行渗流

Half-space last passage percolation with inhomogeneous first row

该研究针对半空间中第一行非齐次的指数最后通行渗流模型，结合TASEP模型，推导了重标最后通行时间的极限分布，并给出半空间点到点最后通行时间的相关关键分析结果。

Weak shape theorem in first passage percolation with infinite passage times

We consider the model of i.i.d. first passage percolation on $\mathbb{Z}^d$ : we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e)<+\infty] >p_c(d)$. Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if $\mathbb{P}[t(e)=0]<p_c(d)$.

首段渗流的单调性

Monotonicity in first-passage percolation

我们考虑$\Z^d$上的标准首过渗流。令$e_1$为第一坐标向量。设$a(n)$为从原点到$ne_1$的预期通过时间。在这篇简短的论文中，我们注意到，$a(n)$是增加的一些强条件下的法律支持的通过时间的边缘。

First-passage percolation on random simple triangulations

We study first-passage percolation on random simple triangulations and their dual maps with independent identically distributed link weights. Our main result shows that the first-passage percolation distance concentrates in an $o_p(n^{1/4})$ window around a constant multiple of the graph distance.

具有非均匀指数时间的梯形图上的首次通过渗流

First-passage percolation on ladder-like graphs with heterogeneous exponential times

我们确定的渐近速度的第一次通过渗流过程中的一些梯形图（或宽度-2拉伸）时，与不同的边缘是独立的，指数分布，但不一定都具有相同的平均值。该方法使用一个特定的马尔可夫链与第一次通过渗流过程和其平稳分布的属性。

Upper Tail Large Deviations in First Passage Percolation

For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance $n$, is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as $\exp (-Θ(n^2))$. However the question of existence of the rate function i.e., whether the log-probability normalized by $n^2$ tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the…

有向最后一次渗流的次线性方差

Sublinear variance for directed last-passage percolation

一系列的第一次通过渗流型模型被认为是证明次线性方差和超扩散的相关属性。我们证明了具有高斯顶点权重的定向最后一次渗流具有次线性方差性质。我们还考虑了其他顶点权重分布。相应的结果得到的基态的“定向聚合物在随机环境中”的模型。

n立方图上Richardson模型的渗流、首次通过渗流和覆盖时间

Percolation, first-passage percolation, and covering times for Richardson's model on the n-cube

研究了具有最近邻边的n立方图B_n ={0，1}^n的边通过概率为p的渗流和首次通过渗流.对于定向和非定向渗流，p=e/n和p=1/n是各自的临界概率。对于具有i.i.d.在原点附近具有密度为1的边缘通过时间，当n->infty时，渗流时间（到达立方体的对角的时间）以概率收敛于1。这解决了大卫·奥尔德斯的一个猜想。当边缘通道分布是标准指数时，未定向边缘的（较小的）渗流时间至少为0.88。这些结果被应用到Richardson的模型（无定向）n立方图。Richardson的模型，也被称为没有恢复的接触过程，将感染的传播建模为连接感染节点和未感染节点的每个边缘上的泊松过程。当n->infty时，覆盖整个n立方图的时间有界于1.41 ~ 14.05之间。

On the number of maximal paths in directed last-passage percolation

We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.

Dual Grothendieck polynomials via last-passage percolation

The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.

Divergence of non-random fluctuation for Euclidean first-passage percolation

The non-random fluctuation is one of the central objects in first passage percolation. It was proved in [Shuta Nakajima. Divergence of non-random fluctuation in First Passage Percolation. {\em Electron. Commun. Probab.} 24 (65), 1-13. 2019.] that for a particular asymptotic direction, it diverges in a lattice first passage percolation with an explicit lower bound. In this paper, we discuss the non-random fluctuation in Euclidean first passage percolations and show that it diverges in dimension $d\geq 2$ in this model also. Compared with the result in \cite{N19}, the present result is proved fo…

首次通过渗流中的测地线

Geodesics in First Passage Percolation

本文考虑了Z^2上一类广泛的遍历首通渗流过程，证明了至少存在四条单侧测地线a.s.我们还证明了在四色Richardson增长模型中共存是可能的。这改善了早期的结果哈格斯特龙和Pemantle，加雷特和Marchand，和霍夫曼谁证明，第一通道渗流至少有两个测地线，共存是可能的两种颜色理查森的增长模型。

Transversal fluctuations for a first passage percolation model

We introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$. In this model, the action of a path depends on the geometry of the path and the travel time. We prove that the transversal fluctuation exponent for point-to-line action minimizers is at least $3/5$.

一个GUE中心极限定理和有向首末道渗流的普适性

A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation

证明了具有有限四阶矩的随机变量的GUE中心极限定理。我们应用这一定理证明，有向的第一次和最后一次通过渗流问题在薄矩形表现出普遍的波动给出的Tracy-Widom法律。

最后一段渗流的黎曼-希尔伯特问题

Riemann-Hilbert problems for last passage percolation

最近三年来，末次通过渗流理论有了新的发展，它在随机置换、随机增长和随机恶性游动等方面有着广泛的应用。结果表明，少数几类模型具有概率分布的行列式公式，可以渐近分析。渐近分析的工具之一是Riemann-Hilbert方法。本文综述了Riemann-Hilbert方法在末道渗流问题中的应用。

有向首段渗流平均值的次高斯收敛速度

Subgaussian rates of convergence of means in directed first passage percolation

我们考虑整数格上的有向首次通过渗流，时间常数为$μ$，从原点到$(n,0,...,0)$的通过时间为$a_{0n}$。结果表明，在一定条件下，通过时间分布，$Ea_{0n} - nμ= O(n^{1/2}(\log\log n)/\log n)$。

末道渗流模型的极限分布

Limiting distribution of last passage percolation models

本文综述了极限分布可求的最后一个渗流模型的一些结果及其应用。

Maximal edge-traversal time in First Passage Percolation

In this paper, we study the maximal edge-traversal time (simply we call maximal weight hereafter) on the optimal paths in the first passage percolation for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth up to a multiplicative constant.

末次通过渗流模型的局部涨落

On the local fluctuations of last-passage percolation models

利用Airy过程描述了Hammersley末次渗流模型的极限涨落，证明了它的局部行为类似于布朗运动.我们的方法是相当简单的，它是基于一定的单调性和良好的控制的平衡测度的Hammersley模型（本地比较）。

连续首达渗流模型中极限形状的可微性

Differentiability of limit shapes in continuous first passage percolation models

本文提出抽象连续作用最小化问题框架，推广连续首达与末达渗流，并据此证明极限形状的可微性，同时给出 Riemannian 模型和基于 Poisson 点的离散时间模型两类实例。

A phase transformation for the number of optimal paths in first passage percolation

We consider the first passage percolation model on the square lattice with an edge weight distribution F. In this paper, we consider the number of optimal paths for two points separated by a long distance. We show that there is a phase transition in the sub-criticality and the criticality.

用修正的Poincare不等式解首达渗流的指数浓度

Exponential concentration for First Passage Percolation through modified Poincare inequalities

我们提供了一个新的指数浓度不等式的首次通过渗流有效的一类广泛的边缘时间分布。这改进并扩展了Benjamini，Kalai和Schramm的结果，该结果给出了Bernoulli边缘时间的方差界。我们的方法是基于一些功能的不平等扩大工作的Rossignol和Falik和Samorodnitsky。

宽度为2的指数链上的首通渗流

First-passage percolation on width-two stretches with exponential link weights

本文研究了顶点集为{1，...，n} \times{0,1}和平移不变边结构。对于六种非平凡情况中的三种，我们通过求解某些递归分布方程并调用遍历论的结果来确定χ是首达时间从（0，0）到（n，0）的期望渐近一步增长，从而得到了渐近渗流速率χ的精确表达式.