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Asymptotics for 2D critical and near-critical first-passage percolation

Abstract

We study Bernoulli first-passage percolation (FPP) on the triangular lattice in which sites have 0 and 1 passage times with probability p and \(1-p\), respectively. Denote by \({\mathcal {C}}_{\infty }\) the infinite cluster with 0-time sites when \(p>p_c\), where \(p_c=1/2\) is the critical probability. Denote by \(T(0,{\mathcal {C}}_{\infty })\) the passage time from the origin 0 to \({\mathcal {C}}_{\infty }\). First we obtain explicit limit theorem for \(T(0,{\mathcal {C}}_{\infty })\) as \(p\searrow p_c\). The proof relies on the limit theorem in the critical case, the critical exponent for correlation length and Kesten’s scaling relations. Next, for the usual point-to-point passage time \(a_{0,n}\) in the critical case, we construct subsequences of sites with different growth rate along the axis. The main tool involves the large deviation estimates on the nesting of CLE\(_6\) loops derived by Miller et al. (Ann Probab 44:1013–1052, 2016). Finally, we apply the limit theorem for critical Bernoulli FPP to a random graph called cluster graph, obtaining explicit strong law of large numbers for graph distance.

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Acknowledgements

The author thanks Geoffrey Grimmett for an invitation to the Statistical Laboratory in Cambridge University, and thanks the hospitality of the Laboratory, where this work was completed. The author also thanks an anonymous referee for a detailed report that contributed to a better presentation of this manuscript. The author was supported by the National Natural Science Foundation of China (No. 11601505), an NSFC Grant No. 11688101 and the Key Laboratory of Random Complex Structures and Data Science, CAS (No. 2008DP173182).

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Appendix: Proof of (55)

Appendix: Proof of (55)

Proof of (55)

The proof is essentially the same as the proof of (2.24) in [14]. Recall that for \(j\ge 0\),

$$\begin{aligned} {\widetilde{\Delta }}_j(\omega ):={\mathbf {E}}_p[T(0,\partial B(L(p)))\mid \widetilde{{\mathscr {F}}}_j]-{\mathbf {E}}_p[T(0,\partial B(L(p)))\mid \widetilde{{\mathscr {F}}}_{j-1}]. \end{aligned}$$

First, let us show that for all \(j\ge 0\),

$$\begin{aligned} {\widetilde{\Delta }}_j(\omega )&=T(\widetilde{{\mathcal {C}}}_{j-1}(\omega ),\widetilde{{\mathcal {C}}}_j(\omega ))(\omega )+{\mathbf {E}}_p'[T(\widetilde{{\mathcal {C}}}_j(\omega ),\partial B(L(p)))(\omega ')]\nonumber \\&\quad -{\mathbf {E}}_p'[T(\widetilde{{\mathcal {C}}}_{j-1}(\omega ),\partial B(L(p)))(\omega ')]. \end{aligned}$$
(86)

For \(j\ge 1\), observe that

$$\begin{aligned} T(0,\partial B(L(p)))=T(0,\widetilde{{\mathcal {C}}}_{j-1})+T(\widetilde{{\mathcal {C}}}_{j-1},\widetilde{{\mathcal {C}}}_j)+T(\widetilde{{\mathcal {C}}}_j,\partial B(L(p))). \end{aligned}$$
(87)

Note that \(T(0,\widetilde{{\mathcal {C}}}_{j-1})\) depends only on the sites in \(\overline{\widetilde{{\mathcal {C}}}_{j-1}}\), and \(T(\widetilde{{\mathcal {C}}}_{j-1},\widetilde{{\mathcal {C}}}_j)\) depends only on the sites in \(\overline{\widetilde{{\mathcal {C}}}_j}\backslash \{\text{ interior } \text{ of } \widetilde{{\mathcal {C}}}_{j-1}\}\). So \(T(0,\widetilde{{\mathcal {C}}}_{j-1})\) is \(\widetilde{{\mathscr {F}}}_{j-1}\)-measurable, and \(T(\widetilde{{\mathcal {C}}}_{j-1},\widetilde{{\mathcal {C}}}_j)\) is \(\widetilde{{\mathscr {F}}}_j\)-measurable. Observe that once \(\widetilde{{\mathcal {C}}}_j\) is fixed, \(T(\widetilde{{\mathcal {C}}}_j,\partial B(L(p)))\) depends only on sites which lie in \(B(L(p))\backslash \overline{\widetilde{{\mathcal {C}}}_j}\); given the configuration of the sites in \(\overline{\widetilde{{\mathcal {C}}}_j}\), the sites in \(B(L(p))\backslash \overline{\widetilde{{\mathcal {C}}}_j}\) are conditionally independent of this configuration. Therefore,

$$\begin{aligned} {\mathbf {E}}_p[T(\widetilde{{\mathcal {C}}}_j,\partial B(L(p)))\mid \widetilde{{\mathscr {F}}}_j](\omega )={\mathbf {E}}_p'[T(\widetilde{{\mathcal {C}}}_j(\omega ),\partial B(L(p)))(\omega ')]. \end{aligned}$$
(88)

This together with (87) and the preceding remarks gives

$$\begin{aligned}&{\mathbf {E}}_p[T(0,\partial B(L(p)))\mid \widetilde{{\mathscr {F}}}_j](\omega ) \nonumber \\&\quad =T(0,\widetilde{{\mathcal {C}}}_{j-1})(\omega )+T(\widetilde{{\mathcal {C}}}_{j-1},\widetilde{{\mathcal {C}}}_j)(\omega )+{\mathbf {E}}_p'[T(\widetilde{{\mathcal {C}}}_j(\omega ),\partial B(L(p)))(\omega ')].\quad \end{aligned}$$
(89)

Similarly, we have

$$\begin{aligned}&{\mathbf {E}}_p[T(0,\partial B(L(p)))\mid \widetilde{{\mathscr {F}}}_{j-1}](\omega )\nonumber \\&\quad =T(0,\widetilde{{\mathcal {C}}}_{j-1})(\omega )+{\mathbf {E}}_p'[T(\widetilde{{\mathcal {C}}}_{j-1}(\omega ),\partial B(L(p)))(\omega ')]. \end{aligned}$$
(90)

Then for \(j\ge 1\), (86) follows by subtracting (90) from (89).

It remains to show (86) for \(j=0\). Observe that

$$\begin{aligned} T(0,\partial B(L(p)))=T(0,\widetilde{{\mathcal {C}}}_1)+T(\widetilde{{\mathcal {C}}}_1,\partial B(L(p))). \end{aligned}$$

Essentially the same proof as above shows that

$$\begin{aligned} {\mathbf {E}}_p[T(0,\partial B(L(p)))\mid \widetilde{{\mathscr {F}}}_1](\omega )=T(0,\widetilde{{\mathcal {C}}}_1)(\omega )+{\mathbf {E}}_p'[T(\widetilde{{\mathcal {C}}}_1(\omega ),\partial B(L(p)))(\omega ')].\nonumber \\ \end{aligned}$$
(91)

It is clear that

$$\begin{aligned} {\mathbf {E}}_p[T(0,\partial B(L(p)))]={\mathbf {E}}_p'[T(0,\partial B(L(p)))]. \end{aligned}$$
(92)

Then for \(j=0\), (86) follows by subtracting (92) from (91).

Finally, note that for \(j\ge 0\),

$$\begin{aligned}&T(\widetilde{{\mathcal {C}}}_j(\omega ),\partial B(L(p)))(\omega ')\nonumber \\&\quad =T(\widetilde{{\mathcal {C}}}_j(\omega ),\widetilde{{\mathcal {C}}}_{l(j,\omega ,\omega ')}(\omega '))(\omega ')+T(\widetilde{{\mathcal {C}}}_{l(j,\omega ,\omega ')}(\omega '),\partial B(L(p)))(\omega '), \end{aligned}$$
(93)
$$\begin{aligned}&T(\widetilde{{\mathcal {C}}}_{j-1}(\omega ),\partial B(L(p)))(\omega ')\nonumber \\&\quad =T(\widetilde{{\mathcal {C}}}_{j-1}(\omega ),\widetilde{{\mathcal {C}}}_{l(j,\omega ,\omega ')}(\omega '))(\omega ')+T(\widetilde{{\mathcal {C}}}_{l(j,\omega ,\omega ')}(\omega '),\partial B(L(p)))(\omega '). \end{aligned}$$
(94)

Substitution of (93) and (94) into the right hand side of (86) yields (55). \(\square \)

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Yao, CL. Asymptotics for 2D critical and near-critical first-passage percolation. Probab. Theory Relat. Fields 175, 975–1019 (2019). https://doi.org/10.1007/s00440-019-00908-2

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Keywords

  • Percolation
  • First passage percolation
  • Correlation length
  • Scaling limit
  • Conformal loop ensemble

Mathematics Subject Classification

  • 60K35
  • 82B43