Phase Diagram of Inhomogeneous Percolation with a Defect Plane

Abstract

Let ${\mathbb {L}}$ be the $d$-dimensional hypercubic lattice and let ${\mathbb {L}}_0$ be an $s$-dimensional sublattice, with $2 \le s < d$. We consider a model of inhomogeneous bond percolation on ${\mathbb {L}}$ at densities $p$ and $\sigma$, in which edges in ${\mathbb {L}}\setminus {\mathbb {L}}_0$ are open with probability $p$, and edges in ${\mathbb {L}}_0$ open with probability $\sigma$. We generalize several classical results of (homogeneous) bond percolation to this inhomogeneous model. The phase diagram of the model is presented, and it is shown that there is a subcritical regime for $\sigma < \sigma ^*(p)$ and $p<p_c(d)$ (where $p_c(d)$ is the critical probability for homogeneous percolation in ${\mathbb {L}}$), a bulk supercritical regime for $p>p_c(d)$, and a surface supercritical regime for $p<p_c(d)$ and $\sigma >\sigma ^*(p)$. We show that $\sigma ^*(p)$ is a strictly decreasing function for $p\in [0,p_c(d)]$, with a jump discontinuity at $p_c(d)$. We extend the Aizenman–Barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase. We prove that the cluster size distribution decays exponentially in the subcritical phase, and sub-exponentially in the supercritical phases. For a model of lattice animals with a defect plane, the free energy is related to functions of the inhomogeneous percolation model, and we show how the percolation transition implies a non-analyticity in the free energy of the animal model. Finally, we present simulation estimates of the critical curve $\sigma ^*(p)$.

Appendix: Differential Inequalities for Inhomogeneous Percolation

In this appendix our aim is to prove the following differential inequalities

$$\begin{aligned}&\mathbf {q} \cdot \nabla \theta ^I (p,\sigma ,\gamma ) \;\le \; 2d\, \chi ^H(p)\,\theta ^I(p,\sigma ,\gamma ) (1-\gamma ) \frac{\partial \theta ^I}{\partial \gamma } , \end{aligned}$$

(95)

$$\begin{aligned}&\theta ^I(p,\sigma ,\gamma ) \;\le \; \gamma \, \frac{\partial \theta ^I}{\partial \gamma } + \left( \theta ^I(p,\sigma ,\gamma )\right) ^2 + \chi ^H(p)\, \theta ^I(p,\sigma ,\gamma ) \left( \mathbf {p}\cdot \nabla \theta ^I(p,\sigma ,\gamma ) \right) \qquad \quad \end{aligned}$$

(96)

where $\nabla = (\frac{\partial }{\partial p}, \frac{\partial }{\partial \sigma })$ and $p\le \sigma$. These are Eqs. (29) and (30).

These inequalities are due to Aizenman and Barsky [1] for homogeneous bond percolation (the proofs can be found in reference [19] as well), and we adapt their proofs to the inhomogeneous model.

Let $B(N)$ be the box of side-length $2N$ centered at the origin of ${\mathbb {L}}$. Denote by $V(N)$ the vertices in $B(N)$. We shall prove the above differential inequalities in $B(N)$, and then take $N\rightarrow \infty$.

Impose periodic boundary conditions on $B(N)$ by identifying its opposite faces. Denote this finite lattice by ${\mathbb {L}}(N)$. Denote the intersection of ${\mathbb {L}}(N)$ and ${\mathbb {L}}_0$ by ${\mathbb {L}}_0(N)$. Denote the set of edges in ${\mathbb {L}}_0(N)$ by ${\mathbb {E}}_0(N)$ and the set of edges in ${\mathbb {L}}(N)$ by ${\mathbb {E}}(N)$.

As before, edges in ${\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)$ are open with the bulk probability $p$, and edges in ${\mathbb {E}}_0(N)$ are open with surface probability $\sigma$.

The open cluster in ${\mathbb {L}}(N)$ at the vertex $x$ is $C_N(x)$, and the open cluster at the origin is $C_N(0) \equiv C_N$. For $y\in V(N)$, define the susceptibility $\chi _N^I (p,\sigma ;y) = E^I_{p,\sigma } |C_N(y)|$.

Introduce the ghost vertex $g$ and edges ${{\mathbb {E}}_g (N) = \{g{\sim }x|\, x\in V(N)\}}$. Edges in ${\mathbb {E}}_g(N)$ are open with probability $\gamma \in (0,1)$. Define $G_N$ to be the collection of vertices in ${\mathbb {L}}(N)$ adjacent to $g$ through open edges in ${\mathbb {E}}_g(N)$.

For inhomogeneous percolation on ${\mathbb {E}}(N) \cup {\mathbb {E}}_g(N)$ with parameters $(p,\sigma ,\gamma )$, let $P^I_{p,\sigma ,\gamma }$ and $E^I_{p,\sigma ,\gamma }$ be the corresponding probability measure and expectation. For $y\in {\mathbb {L}}(N)$, we define the associated quantities

$$\begin{aligned} \theta ^I_N(p,\sigma ,\gamma ;y)&\;=\; P^I_{p,\sigma ,\gamma } (C_N(y)\cap G_N \not = \emptyset )\end{aligned}$$

(97)

$$\begin{aligned} \chi ^I_N(p,\sigma ,\gamma ;y)&\;=\; E^I_{p,\sigma ,\gamma } \left( |C_N (y) |{1}\!\mathrm{l}\{ C_N(y) \cap G_N = \emptyset \}\right) . \end{aligned}$$

(98)

That is, $\theta ^I_N(p,\sigma ,\gamma ;y)$ is the probability that there is an open path from $y$ to $g$. Also write

$$\begin{aligned} \theta ^I_N(p,\sigma ,\gamma ) \,:=\, \theta ^I_N(p,\sigma ,\gamma ;0) \qquad \text{ and }\qquad \chi ^I_N(p,\sigma ,\gamma ) \,:=\, \chi ^I_N(p,\sigma ,\gamma ;0). \end{aligned}$$

Then $\theta _N^I(p,\sigma ,\gamma ) = \theta _N^I(p,\sigma ,\gamma ;y)$ for all $y \in {\mathbb {L}}_0(N)$, and similarly for $\chi ^I_N$. We shall frequently simplify notation by leaving out the arguments when there is no risk of ambiguity, e.g. $\theta _N^I \equiv \theta _N^I(p,\sigma ,\gamma )$. We have

$$\begin{aligned} \theta ^I_N (p,\sigma ,\gamma )&\;=\; 1 - \sum _{n=1}^\infty (1-\gamma )^n\, P^I_{p,\sigma }(|C_N| = n) \qquad \hbox {and}\\ \chi ^I_N (p,\sigma ,\gamma )&\;=\; \sum _{n=1}^\infty n\,(1-\gamma )^n P^I_{p,\sigma } (|C_N|=n). \end{aligned}$$

We immediately obtain the following analogue of Eq. (27):

$$\begin{aligned} \chi ^I_N (p,\sigma ,\gamma ) = (1-\gamma ) \frac{\partial \theta ^I_N}{\partial \gamma }. \end{aligned}$$

(99)

The functions $\theta ^I (p,\sigma ,\gamma )$ and $\chi ^I (p,\sigma ,\gamma )$ are defined in the usual way for the infinite lattice ${\mathbb {L}}$ with the ghost vertex $g$ and edges ${\mathbb {E}}_g$, and with ${\mathbb {L}}_0$ as defined before:

$$\begin{aligned} \theta ^I(p,\sigma ,\gamma )&\;=\; \theta ^I(p,\sigma ,\gamma ;0)\;=\; P^I_{p,\sigma ,\gamma } (|C| = \infty ) \\ \chi ^I(p,\sigma ,\gamma )&\;=\; \chi ^I(p,\sigma ,\gamma ;0) \;=\; E^I_{p,\sigma ,\gamma }(|C|{1}\!\mathrm{l}\{C\cap G=\emptyset \}) \end{aligned}$$

where $C$ is the cluster at the origin. Note that for $\gamma >0$, $C \cap G$ is not empty with probability one when $|C|=\infty$.

The proof of the following lemma is similar to the proof for homogeneous percolation in appendix I of reference [19].

Lemma 6

For all $\gamma \in (0,1)$ and $p,\sigma \in (0,1)$,

$$\begin{aligned} \lim _{N\rightarrow \infty } \theta ^I_N (p,\sigma , \gamma ) \;=\; \theta ^I(p,\sigma ,\gamma ), \end{aligned}$$

and similarly,

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{\partial \theta ^I_N}{\partial p} \;=\; \frac{\partial \theta ^I}{\partial p},\quad \lim _{N\rightarrow \infty } \frac{\partial \theta ^I_N}{\partial \sigma } \;=\; \frac{\partial \theta ^I}{\partial \sigma },\,\;\hbox {and}\,\; \lim _{N\rightarrow \infty } \frac{\partial \theta ^I_N}{\partial \gamma } \;=\; \frac{\partial \theta ^I}{\partial \gamma }.\end{aligned}$$

$\square$

By Eqs. (27) and (99) and Lemma 6, we have

$$\begin{aligned} \lim _{N\rightarrow \infty } \chi ^I_N (p,\sigma ,\gamma ) \;=\; \chi ^I(p,\sigma ,\gamma ). \end{aligned}$$

(100)

1.1 Uniform Bounds for Vertex-Dependent Functions

Let $P^H_{p,\gamma }$ and $E^H_{p,\gamma }$ be the probability measure and expectation for homogeneous percolation with a ghost field. Let $\chi _N^H(p,\gamma ) = E^H_{p,\gamma } (|C_N|{1}\!\mathrm{l}\{ C_N\cap G_N = \emptyset \})$ be the corresponding susceptibility in $B(N)$. Also let $\chi _N^H(p) = \chi ^H_N(p,0)$.

Proving the differential inequalities requires the following useful lemma.

Lemma 7

Assume $p\le \sigma$ and $y\in B(N)$. Then

• (a) $\theta _N^I (p,\sigma ,\gamma ;y) \;\le \; \chi _N^H(p) \, \theta _N^I (p,\sigma ,\gamma )$ for every $\gamma \in (0,1)$;

• (b) $\chi _N^I(p,\sigma ;y) \;\le \; \chi _N^H(p) \,\chi _N^I(p,\sigma ;0)$ (where $\chi _N^I(p,\sigma ;y):=\chi _N^I(p,\sigma ,0;y)$).

Proof

Since $\chi _N^H(p) \ge 1$, it is only necessary to consider the case that $y\not \in {\mathbb {L}}_0(N)$.

(a) Suppose that $C_N(y) \cap G_N \not = \emptyset$. Then there is an open path from $y$ to a point of $G_N$. This path either uses no edge of ${\mathbb {E}}_0(N)$, or there exists an open self-avoiding path $\pi$ from $y$ to a vertex of $G_N$ which passes through an edge of ${\mathbb {E}}_0(N)$. In the latter case, let $z$ be the earliest point of $\pi$ that is an endpoint of a bond in $\pi \cap {\mathbb {L}}_0(N)$. Then $z\in {\mathbb {L}}_0(N)$, and the part of $\pi$ from $y$ to $z$ is disjoint from the part of the path from $z$ to a vertex of $G_N$.

We formalize the above as follows. For given fixed $y$ and for each $z\in {\mathbb {L}}_0(N)$, define the events

• $\widetilde{A}_N$ is the event that there is an open path from $y$ to $G_N$ in ${\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)$;

• $\widetilde{D}_N(z)$ is the event that there is an open path from $y$ to $z$ in ${\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)$;

• $D^*_N(z)$ is the event that there is an open path from $z$ to $G_N$ in $B(N)$.

Observe that

$$\begin{aligned} P^I_{p,\sigma ,\gamma } (\widetilde{A}_N) \;=\; P_{p,\gamma }^H(\widetilde{A}_N) \; \le \; \theta _N^I(p,p,\gamma ) \;\le \; \theta _N^I(p,\sigma ,\gamma ) \qquad \hbox {(since }p\le \sigma ). \end{aligned}$$

(101)

Using standard percolation notation, $\widetilde{D}_N(z) \circ D^*_N(z)$ is the event that $\widetilde{D}_N(z)$ and $D^*_N(z)$ occur disjointly—that is, there exist two disjoint sets of open edges such that the first set guarantees occurrence of $\widetilde{D}_N(z)$ and the second set guarantees occurrence of $D^*_N(z)$.

We then observe that

$$\begin{aligned} \left\{ C_N(y) \cap G_N \not = \emptyset \right\} \subset \widetilde{A}_N \cup \bigcup _{z\in {\mathbb {L}}_0(N)} \left( \widetilde{D}_N(z) \circ D^*_N(z) \right) \end{aligned}$$

(102)

since occurrence of of the event $\left\{ C_N(y) \cap G_N \not = \emptyset \right\}$ implies that either $\widetilde{A}_N$ occurs or $\left( \widetilde{D}_N(z) \circ D^*_N(z) \right)$ occurs for some $z\in {\mathbb {L}}_0(N)$.

From Eqs. (102) and (101) and the BK Inequality [19], we see that

$$\begin{aligned}&P^I_{p,\sigma ,\gamma }\left( C_N(y) \cap G_N \not = \emptyset \right) \\&\;\le \; P^I_{p,\sigma ,\gamma } \left( \widetilde{A}_N \right) \;+ \sum _{z\in {\mathbb {L}}_0(N)} P^I_{p,\sigma ,\gamma }\left( \widetilde{D}_N(z) \circ D^*_N(z) \right) \\&\;\le \; \theta ^I_N (p,\sigma ,\gamma )\;+ \sum _{z\in {\mathbb {L}}_0(N)}P^I_{p,\sigma ,\gamma } ( \widetilde{D}_N(z) ) \, P^I_{p,\sigma ,\gamma } ( D^*_N(z) ) \\&\;\le \; \theta ^I_N (p,\sigma ,\gamma ) \;+ \sum _{z\in {\mathbb {L}}_0(N)} P^H_{p} ( z\in C_N(y) ) \, P^I_{p,\sigma ,\gamma } ( C_N(z) \cap G_N \not =\emptyset ) \\&\;=\; \left( 1 \;+ \sum _{z\in {\mathbb {L}}_0(N)} P^H_{p} ( z\in C_N(y) ) \right) \theta ^I_N (p,\sigma ,\gamma ) \\&\;\le \; \chi _N^H(p)\, \theta ^I_N (p,\sigma ,\gamma ). \end{aligned}$$

(b) Fix $y\in {\mathbb {L}}(N)\setminus {\mathbb {L}}_0(N)$. Let $\widetilde{D}_N(z)$ be defined as in part (a). For $w\in B(N)$, let $\left\{ y \leftrightarrow _N w \right\}$ denote the event that $y$ and $w$ are connected in $B(N)$ by a path of open edges.

Similarly to the proof of part (a), we see that for each $w\in B(N)$

$$\begin{aligned} P^I_{p,\sigma }\left( y \leftrightarrow _N w \right)&\;\le \; P^I_{p,\sigma } \left( \widetilde{D}_N (w) \right) \; + \sum _{z\in {\mathbb {L}}_0(N)} P^I_{p,\sigma } \left( \widetilde{D}_N(z) \circ \left\{ z\leftrightarrow _N w\right\} \right) \\&\;\le \; P^I_{p,\sigma } \left( \widetilde{D}_N (w) \right) \; + \sum _{z\in {\mathbb {L}}_0(N)} P^I_{p,\sigma } \left( \widetilde{D}_N(z) \right) \, P^I_{p,\sigma } \left( \left\{ z\leftrightarrow _N w\right\} \right) \nonumber \\&\;\le \; P^H_{p} \! \left( \left\{ y \leftrightarrow _N w \right\} \right) \, + \!\!\! \sum _{z\in {\mathbb {L}}_0(N)} \! P^H_{p} \left( \left\{ y \leftrightarrow _N z \right\} \right) \,\! P^I_{p,\sigma } \left( \left\{ z\leftrightarrow _N w\right\} \right) . \nonumber \end{aligned}$$

(103)

Since $p\le \sigma$, we have $\chi _N^H(p) \le \chi _N^I(p,\sigma ;0) = \chi ^I_N(p,\sigma ;z)$ for every $z\in {\mathbb {L}}_0(N)$. Using this and summing Eq. (103) over $w$, we obtain

$$\begin{aligned}&\chi _N^I(p,\sigma ;y) \;=\; \sum _{w\in B(N)} P^I_{p,\sigma }\left( y \leftrightarrow _N w \right) \\&\;\le \; \sum _{w\in B(N)} P^H_{p} \! \left( \left\{ y \leftrightarrow _N w \right\} \right) + \sum _{w\in B(N)} \sum _{z\in {\mathbb {L}}_0(N)} P^H_{p} \left( \left\{ y \leftrightarrow _N z \right\} \right) \, P^I_{p,\sigma } \left( \left\{ z\leftrightarrow _N w\right\} \right) \\&\;=\;\chi _N^H(p) \;+ \sum _{z\in {\mathbb {L}}_0(N)} P^H_{p} \left( \left\{ y \leftrightarrow _N z \right\} \right) \, \chi _N^I(p,\sigma ;z) \\&\le \left( 1+ \sum _{z\in {\mathbb {L}}_0(N)} P^H_{p} \left( \left\{ y \leftrightarrow _N z \right\} \right) \right) \chi _N^I(p,\sigma ;0) \;\le \; \chi _N^H (p)\,\chi ^I_N(p,\sigma ;0). \end{aligned}$$

This completes the proof. $\square$

1.2 The First Differential Inequality

The first differential inequality is defined in terms of $\theta _N^I \equiv \theta ^I_N(p,\sigma ,\gamma )$ as follows.

Lemma 8

For $p,\sigma ,\gamma \in (0,1)$, let $\mathbf {p} = (p,\sigma )$ and $\nabla \equiv \left( \frac{\partial }{\partial p}, \frac{\partial }{\partial \sigma }\right)$. Define $\mathbf {q} = \mathbf {1}-\mathbf {p} = (1-p,1-\sigma )$. If $p \le \sigma$, then we have

$$\begin{aligned} \mathbf {q} \cdot \nabla \theta _N^I \;\le \; 2d\, \chi ^H_N(p)\, (1-\gamma ) \,\theta ^I_N\,\frac{\partial \theta _N^I}{\partial \gamma }. \end{aligned}$$

Proof

The proof is similar to the proof for homogeneous percolation (see for example reference [19]) and proceeds by applying Russo’s formula to the event $\{ C_N \cap G_N \not = \emptyset \}$, conditioned on $G_N$.

Let $\Gamma$ be a realisation of $G_N$, i.e. a subset of vertices of $B(N)$. The event $A_N (\Gamma ) = \{ C_N \cap \Gamma \not =\emptyset \}$ is increasing. Hence by Russo’s formula [19],

$$\begin{aligned} \frac{\partial }{\partial p} P_{p,\sigma }^I (A_N(\Gamma ))&\;=\; \sum _{e\in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)} P_{p,\sigma }^I (e\hbox { is pivotal for }A_N(\Gamma )), \end{aligned}$$

(104)

$$\begin{aligned} \frac{\partial }{\partial \sigma } P_{p,\sigma }^I (A_N(\Gamma ))&\;=\; \sum _{e\in {\mathbb {E}}_0(N)} P_{p,\sigma }^I (e\hbox { is pivotal for }A_N(\Gamma )). \end{aligned}$$

(105)

First consider Eq. (104). An edge $e=x{\sim }y$ is pivotal for $A_N (\Gamma )$ if and only if the following all occur in ${\mathbb {E}}(N) \setminus \{e\}$: (1) there is no open path from the origin to $\Gamma$, (2) exactly one of $x$ and $y$ is joined to the origin by an open path, and (3) the other vertex is joined to a vertex of $\Gamma$ by an open path. Hence,

$$\begin{aligned}&(1-p)\frac{\partial }{\partial p} P_{p,\sigma }^I (A_N(\Gamma )) \\ \;=\;&\sum _{e\in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)} P_{p,\sigma }^I (e\hbox { is closed}) P_{p,\sigma }^I (e\hbox { is pivotal for }A_N(\Gamma )) \\ \;=\;&\sum _{x{\sim }y\in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)} P_{p,\sigma }^I (x\in C_N, C_N \cap \Gamma = \emptyset , C_N(y)\cap \Gamma \not =\emptyset ), \end{aligned}$$

where the last summation is over all ordered pairs $(x,y)$ of vertices such that the (undirected) edge $x{\sim }y$ is in ${\mathbb {E}}(N) \setminus {\mathbb {E}}_0(N)$. Put $q=1-p$ and average the left hand side of the above over $\Gamma$:

$$\begin{aligned}&q \sum _{\Gamma } P_{p,\sigma ,\gamma }^I (G_N = \Gamma )\, \frac{\partial }{\partial p} P_{p,\sigma }^I (A_N(\Gamma )) \\&\quad = q\frac{\partial }{\partial p} \left[ \sum _{\Gamma } P_{p,\sigma ,\gamma }^I(C_N \cap \Gamma \not =\emptyset )\, P_{p,\sigma ,\gamma }^I(G_N=\Gamma ) \right] \\&\quad = q\frac{\partial }{\partial p} P_{p,\sigma ,\gamma }^I(C_N \cap G_N \not =\emptyset ) \\&\quad = q\frac{\partial }{\partial p} \theta _N^I(p,\sigma ,\gamma ). \end{aligned}$$

Here it is important that the sum over $\Gamma$ has a finite number of terms.

This shows that

$$\begin{aligned} q\frac{\partial }{\partial p} \theta _N^I&\;=\; \sum _{x{\sim }y\in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)}P_{p,\sigma ,\gamma }^I (x\in C_N, C_N \cap G_N = \emptyset , C_N(y)\cap G_N \not =\emptyset ). \end{aligned}$$

Observe that $C_N$ and $C_N(y)$ must be disjoint on the right hand side.

Exactly the same set of arguments applied to Eq. (105) gives (with $\tau = 1-\sigma$)

$$\begin{aligned} \tau \frac{\partial }{\partial \sigma } \theta _N^I \;=\;\sum _{x{\sim }y\in {\mathbb {E}}_0(N)} P_{p,\sigma ,\gamma }^I (x\in C_N, C_N\cap G_N = \emptyset , C_N(y)\cap G_N \not =\emptyset ). \end{aligned}$$

(106)

Adding the last two equations together then produces

$$\begin{aligned} \mathbf {q} \cdot \nabla \theta _N^I\;=\; \sum _{x{\sim }y\in {\mathbb {E}}(N)} P_{p,\sigma ,\gamma }^I(x\in C_N, C_N \cap G_N = \emptyset , C_N(y)\cap G_N \not =\emptyset ). \end{aligned}$$

(107)

The right hand side of Eq. (107) must be bounded next. This is done by conditioning on the cluster at the origin. The last equation becomes

$$\begin{aligned}&\mathbf {q} \cdot \nabla \theta _N^I\\&\;=\; \sum _{x{\sim }y}\left[ \sum _{\Xi }P_{p,\sigma }^I(C_N = \Xi )\,P_{p,\sigma ,\gamma }^I(C_N \cap G_N = \emptyset , C_N(y)\cap G_N \not =\emptyset {\,\biggr |\,}C_N = \Xi )\right] \nonumber \end{aligned}$$

(108)

where the outer sum is over ordered pairs $(x,y)$ such that $x{\sim }y\in {\mathbb {E}}(N)$, and the inner sum is over all connected graphs $\Xi$ containing $\{0,x\}$ and not containing $y$.

Conditioned on $C_N=\Xi$, the events $C_N \cap G_N = \emptyset$ and $C_N(y)\cap G_N \not =\emptyset$ are independent (the first depends only on vertices of $\Xi$, and the second depends only on vertices and edges outside $\Xi$). Hence

$$\begin{aligned}&P_{p,\sigma ,\gamma }^I \left( C_N \cap G_N = \emptyset ,C_N(y)\cap G_N \not =\emptyset {\,\biggr |\,}C_N = \Xi \right) \\&\qquad \;=\; P_{p,\sigma ,\gamma }^I\biggl (C_N \cap G_N = \emptyset {\,\biggr |\,}C_N = \Xi \biggr ) \,P_{p,\sigma ,\gamma }^I \biggr (C_N(y) \cap G_N \not =\emptyset {\,\biggr |\,}C_N = \Xi \biggr ). \end{aligned}$$

The condition $C_N = \Xi$ in the last factor restricts the set of possible open paths from $y$ to a vertex in $G_N$ (since $y\not \in C_N$). Hence

$$\begin{aligned} P_{p,\sigma ,\gamma }^I (C_N(y) \cap G_N \not =\emptyset \big | C_N = \Xi ) \;\le \; P_{p,\sigma ,\gamma }^I (C_N(y) \cap G_N \not =\emptyset ) \;=\; \theta _N^I(p,\sigma ,\gamma ; y). \end{aligned}$$

This shows that

$$\begin{aligned}&\mathbf {q} \cdot \nabla \theta _N^I\nonumber \\&\quad \le \; \sum _{x{\sim }y}\left[ \sum _{\Xi }P_{p,\sigma }^I(C_N = \Xi )\, P_{p,\sigma ,\gamma }^I (C_N \cap G_N = \emptyset | C_N = \Xi )\, \theta _N^I(p,\sigma ,\gamma ; y) \right] \nonumber \\&\quad =\; \sum _{x{\sim }y}\left[ P_{p,\sigma ,\gamma }^I (x\in C_N, y\not \in C_N, C_N \cap G_N = \emptyset ) \, \theta _N^I(p,\sigma ,\gamma ; y) \right] \nonumber \\&\quad \le \; \chi ^H_N(p)\,\theta _N^I(p,\sigma ,\gamma ) \sum _{x{\sim } y} P_{p,\sigma ,\gamma }^I (x\in C_N, y\not \in C_N, C_N \cap G_N = \emptyset ) \end{aligned}$$

(109)

where the final inequality comes from Lemma 7(a). It remains to bound the last summation.

$$\begin{aligned}&\sum _{x{\sim } y}P_{p,\sigma ,\gamma }^I (x\in C_N, y\not \in C_N,C_N \cap G_N = \emptyset )\\&\quad \le 2d \sum _{x} P_{p,\sigma ,\gamma }^I (x\in C_N, C_N \cap G_N = \emptyset ) \\&\quad = 2d\, E_{p,\sigma ,\gamma }^I \left( \left| C_N \right| \,{1}\!\mathrm{l}\{C_N \cap G_N = \emptyset \} \right) \\&\quad = 2d\, \chi _N^I(p,\sigma ,\gamma ) = 2d\,(1-\gamma ) \frac{\partial \theta _N^I}{\partial \gamma } \end{aligned}$$

by Eqs. (98) and (99). Putting this all together then gives the desired inequality. $\square$

1.3 The Second Differential Inequality

The second differential inequality is the following (again writing $\theta _N^I$ for $\theta _N^I(p,\sigma ,\gamma )$).

Lemma 9

For $p,\sigma ,\gamma \in (0,1)$ let $\mathbf {p} = (p,\sigma )$ and $\nabla \equiv \left( \frac{\partial }{\partial p}, \frac{\partial }{\partial \sigma }\right)$. If $p \le \sigma$, then

$$\begin{aligned} \theta _N^I \;\le \; \gamma \, \frac{\partial \theta _N^I}{\partial \gamma } + \left( \theta _N^I\right) ^2 + \chi _N^H(p)\, \theta _N^I \left( \mathbf {p}\cdot \nabla \theta _N^I \right) . \end{aligned}$$

Proof

Observe that

$$\begin{aligned} \theta _N^I(p,\sigma ,\gamma )\;=\;&P_{p,\sigma ,\gamma }^I(C_N \cap G_N \not =\emptyset ) \nonumber \\ \;=\;&P_{p,\sigma ,\gamma }^I(\left| C_N \cap G_N \right| = 1)+ P_{p,\sigma ,\gamma }^I(\left| C_N \cap G_N \right| \ge 2). \end{aligned}$$

(110)

The first term in Eq. (110) can calculated:

$$\begin{aligned} P_{p,\sigma ,\gamma }^I(\left| C_N \cap G_N \right| = 1)\;=\;&\sum _{n=1}^\infty n\gamma (1-\gamma )^{n-1}\,P_{p,\sigma ,\gamma }^I(\left| C_N \right| =n) \nonumber \\ \;=\;&\frac{\gamma \,\chi _N^I(p,\sigma ,\gamma )}{1-\gamma } \nonumber \\ \;=\;&\gamma \, \frac{\partial \theta _N^I}{\partial \gamma } \end{aligned}$$

(111)

by Eq. (99).

It remains to bound the second term in (110). Define the event

$$\begin{aligned} A_x \;=\; \{ x\in G_N\hbox { or }x\hbox { is joined to }G_N\hbox { by an open path} \}. \end{aligned}$$

Then $A_x \circ A_x$ is the event that there exist two distinct vertices $v_1$ and $v_2$ in $G_N$ and two edge-disjoint open paths joining these vertices to $x$. If $x\in G_N$, then one these paths may be the singleton $x$.

It follows that

$$\begin{aligned} P_{p,\sigma ,\gamma }^I (\left| C_N \cap G_N \right| \ge 2)&= P_{p,\sigma ,\gamma }^I (A_0\circ A_0) \\&+\, P_{p,\sigma ,\gamma }^I (\left| C_N \cap G_N \right| \;\ge \; 2,\hbox { and }A_0\circ A_0\hbox { does not occur}). \nonumber \end{aligned}$$

(112)

By the BK inequality,

$$\begin{aligned} P_{p,\sigma ,\gamma }^I (A_0\circ A_0) \;\le \;\left( P_{p,\sigma ,\gamma }^I (A_0) \right) ^2 \;=\; \left( \theta _N^I(p,\sigma ,\gamma )\right) ^2. \end{aligned}$$

(113)

The remaining term is the probability that $\left| C_N \cap G_N \right| \ge 2$ but there do not exist two edge-disjoint paths from the origin to distinct vertices in $G_N$. If this occurs, then there exists an edge $x{\sim }y$ in ${\mathbb {E}}(N)$ with the following properties:

• $x{\sim }y$ is open;

• If $x{\sim }y$ is deleted in ${\mathbb {L}}(N)$, then three events occur:

  • 1. there is no open path from the origin to a vertex of $G_N$;

  • 2. $x$ is joined to the origin by an open path;

  • 3. the event $A_y\circ A_y$ occurs.

The probability that a particular edge $x{\sim }y$ has these properties is

$$\begin{aligned} pq^{-1}\, P_{p,\sigma ,\gamma }^I(x{\sim }y\hbox { is closed, }x\in C_N, C_N\cap G_N =\emptyset , A_y\circ A_y) \end{aligned}$$

if $x{\sim }y \in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)$; if $x{\sim }y \in {\mathbb {E}}_0(N)$, then we get the above expression with $\sigma (1-\sigma )^{-1}$ instead of $pq^{-1}$. Therefore we obtain the bound

$$\begin{aligned} P_{p,\sigma ,\gamma }^I (\left| C_N \cap G_N \right| \;\ge \; 2,\hbox { and } A_0\circ A_0\hbox { does not occur}) \;\le \; S_1\,+\,S_2, \end{aligned}$$

where

$$\begin{aligned} S_1 \,=\, pq^{-1}\, \sum _{x{\sim }y\in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)}P_{p,\sigma ,\gamma }^I(x\in C_N, C_N\cap G_N =\emptyset , A_y\circ A_y) \end{aligned}$$

(114)

$$\begin{aligned} \text{ and }\qquad S_2 \,=\,\sigma \tau ^{-1}\, \sum _{x{\sim }y\in {\mathbb {E}}_0(N)}P_{p,\sigma ,\gamma }^I(x\in C_N, C_N\cap G_N =\emptyset , A_y\circ A_y), \end{aligned}$$

(115)

writing $\tau = 1-\sigma$.

Consider a summand from Eq. (114) and (115) conditioned on $C_N=\Xi$, with $x\in \Xi$ and $y\not \in \Xi$. Using conditional independence of the events $A_y$ and $\{ C_N \cap G_N = \emptyset \}$, and the BK inequality, we obtain

$$\begin{aligned} P_{p,\sigma ,\gamma }^I&\left( x\in C_N, C_N\cap G_N =\emptyset , A_y\circ A_y{\,\biggr |\,}C_N=\Xi \right) \\&\;=\; P_{p,\sigma ,\gamma }^I\left( C_N\cap G_N =\emptyset {\,\biggr |\,}C_N=\Xi \right) \, P_{p,\sigma ,\gamma }^I\left( A_y\circ A_y{\,\biggr |\,}C_N=\Xi \right) \\&\;\le \; P_{p,\sigma ,\gamma }^I\left( C_N\cap G_N =\emptyset {\,\biggr |\,}C_N=\Xi \right) \,\left( P_{p,\sigma ,\gamma }^I (A_y {\,\biggr |\,}C_N=\Xi ) \right) ^2 \\&\;\le \; P_{p,\sigma ,\gamma }^I \left( C_N\cap G_N =\emptyset {\,\biggr |\,}C_N=\Xi \right) \,P_{p,\sigma ,\gamma }^I \left( A_y {\,\biggr |\,}C_N=\Xi \right) \,P_{p,\sigma ,\gamma }^I (A_y) \\&\;=\; P_{p,\sigma ,\gamma }^I\left( C_N\cap G_N =\emptyset , A_y{\,\biggr |\,}C_N=\Xi \right) \,\theta _N^I(p,\sigma ,\gamma ;y). \end{aligned}$$

Substitute this into Eq. (114) and average over $\Xi$. This gives the upper bound

$$\begin{aligned} S_1 \;\le \;pq^{-1}\, \sum _{x{\sim }y \in {\mathbb {E}}(N)\setminus {\mathbb {E}}_0(N)}P_{p,\sigma ,\gamma }^I\left( x\in C_N, C_N\cap G_N =\emptyset , A_y\right) \,\theta _N^I(p,\sigma ,\gamma ;y). \end{aligned}$$

Next, by Eq. (106) and Lemma 7(a), we obtain (with $\theta _N^I(p,\sigma ,\gamma )\equiv \theta _N^I$)

$$\begin{aligned} S_1 \;\le \; pq^{-1}\, \left( q\, \frac{\partial \theta _N^I }{\partial p}\right) \, \chi _N^H(p) \, \theta _N^I. \end{aligned}$$

The analogous bound for Eq. (115) is

$$\begin{aligned} S_2 \;\le \; \sigma \tau ^{-1}\, \left( \tau \, \frac{\partial \theta _N^I }{\partial \sigma } \right) \, \chi _N^H(p) \, \theta _N^I. \end{aligned}$$

Hence,

$$\begin{aligned} P_{p,\sigma ,\gamma }^I&(\left| C_N \cap G_N \right| \ge 2, \hbox { and } A_0\circ A_0\hbox { does not occur}) \\&\;\le \; \left( p \,\frac{\partial \theta _N^I }{\partial p} + \sigma \,\frac{\partial \theta _N^I }{\partial \sigma } \right) \, \chi _N^H(p) \, \theta _N^I. \end{aligned}$$

Putting this together with Eqs. (110), (112), (111) and (113) completes the proof of the desired inequality. $\square$

1.4 The Final Differential Inequalities

To complete the proof of the two differential inequalities (29) and (30), we take the $N\rightarrow \infty$ limit in Lemmas 8 and 9. Using Lemma 6 and Eq. (100), the result is the following theorem.

Theorem 8

For $p,\sigma ,\gamma \in (0,1)$, write $\mathbf {p} = (p,\sigma )$, $\nabla \equiv \left( \frac{\partial }{\partial p}, \frac{\partial }{\partial \sigma }\right)$, $\mathbf {q} = \mathbf {1}-\mathbf {p} = (1{-}p,1{-}\sigma )$, and $\theta ^I \equiv \theta ^I(p,\sigma ,\gamma )$. If $p \le \sigma$, then

$$\begin{aligned} \mathbf {q} \cdot \nabla \theta ^I \;\;&\le \;\; 2d\, (1-\gamma ) \,\chi ^H(p)\, \theta ^I \,\frac{\partial \theta ^I}{\partial \gamma } \qquad \qquad \hbox {and} \\ \theta ^I&\le \;\; \gamma \, \frac{\partial \theta ^I}{\partial \gamma }+ \left( \theta ^I\right) ^2+ \chi ^H(p)\, \theta ^I\left( \mathbf {p}\cdot \nabla \theta ^I \right) . \end{aligned}$$