The model and the theorem

Let \(d\ge3\) be an integer, let \(e_1,\ldots,e_d\) be the coordinate basis, and set \[U=\{\pm e_1,\ldots,\pm e_d\},\qquad \mathcal P_U=\left\{p\in[0,1]^U:\sum_{e\in U}p(e)=1\right\}, \qquad \Omega=\mathcal P_U^{\mathbb Z^d}.\] Equip \(\Omega\) with its product Borel \(\sigma\)-field. An environment \(\omega\in\Omega\) assigns a transition vector \(\omega(x,\cdot)\) to each site \(x\). We call this vector the row at \(x\). Let \(\mu\) be a probability measure on \(\mathcal P_U\), and let \(\mathbb P=\mu^{\otimes\mathbb Z^d}\). Thus the rows are independent and identically distributed; entries within one row need not be independent. Assume only \[ \mu\bigl(\{p\in\mathcal P_U:p(e)>0\text{ for every }e\in U\}\bigr)=1. \tag{1}\] We call this strict ellipticity. By countability, it gives \(\omega(x,e)>0\) simultaneously for all sites and steps, almost surely. Unlike uniform ellipticity, it permits the smallest row entry to be arbitrarily close to zero.

The path space is \(\mathcal X=(\mathbb Z^d)^{\mathbb N_0}\) with coordinate maps \(X_n\) and the \(\sigma\)-field \(\sigma(X_n:n\ge0)\). For \(x\in\mathbb Z^d\) and a fixed environment, the quenched law \(P_{x,\omega}\) is the Markov-chain law specified by \[P_{x,\omega}(X_0=x)=1,\qquad P_{x,\omega}(X_{n+1}=X_n+e\mid X_0,\ldots,X_n)=\omega(X_n,e).\] The annealed law and its expectation are \[P_x(B)=\int P_{x,\omega}(B)\,\mathbb P(d\omega),\qquad E_x.\] When both environment and path are needed, we use the joint law \(\mathsf P_x(d\omega,dX)=\mathbb P(d\omega)P_{x,\omega}(dX)\). The path filtration is \(\mathcal F_n=\sigma(X_0,\ldots,X_n)\); on the joint space the larger filtration \(\mathcal G_n=\sigma(\omega,X_0,\ldots,X_n)\) also reveals the entire environment. Quenched Markov arguments use \(\mathcal G_n\). The endpoint-posterior martingales later in the proof use \(\mathcal F_n\), with the environment integrated out. An unconditioned annealed path law will also be called the raw law.

For a nonzero vector \(\ell\in\mathbb R^d\), define \[A_\ell=\{X_n\cdot\ell\longrightarrow+\infty\}.\]

The direction is any fixed real vector; no rationality assumption is made, and no assertion of one exceptional null set uniform over all directions is needed. No inverse-transition or logarithmic moment, drift, balance, reversibility, or speed condition supplements (1). In the proof, a finite first moment for a spatial radius is derived under the hypothesis that both signs have positive probability. No first moment for a regeneration duration, positive speed or ballisticity assertion is used or obtained from that estimate.

Prior work

The directional question goes back to Kalikow (Kalikow 1981); see also the historical discussion in (Zerner 2007). The classical sign-union zero–one law concerns \(P_0(A_\ell\cup A_{-\ell})\), not \(P_0(A_\ell)\). It originates in the uniformly elliptic theory of Kalikow (Kalikow 1981); its strict-ellipticity extension is (Zerner and Merkl 2001, Proposition 3), as recalled in (Zerner 2007, Equation (1)). Zerner and Merkl proved the one-sign directional law in dimension two for iid strictly elliptic environments (Zerner and Merkl 2001, Theorem 1), and Zerner gave a shorter proof (Zerner 2007). Slonim extended the planar iid law to bounded jumps when almost surely there is a unique infinite communicating class reachable from every site (Slonim 2024, Theorem 2.1). Theorem 1 establishes the directional zero–one conjecture for nearest-neighbor walks in every higher dimension under strict ellipticity.

The difficulty in higher dimensions is that the planar arguments force oppositely directed paths to meet by geometry; their quantitative counterpart must be supplied differently here. Berger proved that in dimensions \(d\ge5\) an iid uniformly elliptic walk has at most one nonzero limiting velocity (Berger 2008, Theorem 1.1). This allows a second velocity equal to zero and leaves open the possibility of opposite directional escape at zero speed. Independence is also substantive: Bramson, Zeitouni, and Zerner constructed stationary, polynomially mixing, uniformly elliptic environments in \(d\ge3\) with positive probabilities of linear escape in opposite directions (Bramson et al. 2006, Theorem 3). Their Open Problem 3 formulates the higher-dimensional question for iid uniformly elliptic environments.

A non-uniformly elliptic special case is provided by Dirichlet-distributed rows. Sabot and Tournier obtained positive-probability transience in coordinate directions with positive corresponding mean drift, using reversal identities specific to that distribution (Sabot and Tournier 2011, Theorem 1). Bouchet proved the fixed-direction zero–one law in this class for \(d\ge3\) by studying an accelerated walk and an invariant law for the environment viewed from the particle (Bouchet 2013, Corollary 4). Tournier identified the asymptotic direction as that of the initial drift when this drift is nonzero (Tournier 2015, Corollary 1). The argument below applies to arbitrary iid strictly positive rows and uses only translations of its path pieces.

The regeneration structure of Sznitman and Zerner (Sznitman and Zerner 1999) is a central tool. We give the factorization and width arguments needed below, including the distinction between real projected heights and integer record indices. Simenhaus proved that distinct deterministic asymptotic directions occurring with positive probability must be opposite (Simenhaus 2007, Proposition 1). His existence theorem for an asymptotic direction assumes transience on a nonempty open set of directions. Drewitz and Ramírez obtained a corresponding two-ray description under sign-union transience on an open set, equivalently in \(d\) linearly independent directions (Drewitz and Ramírez 2010, Theorem 1.8). Those hypotheses do not follow here from the initial assumption in one direction. In our contradiction argument, the spatial radius estimate supplies the two rays first; a self-contained argument then proves their antipodality. The remaining estimates concern contacts between two arrangements of regeneration pieces.

Proof strategy

Two fresh-row arguments make the proof work without a common positive lower bound on the transition probabilities. First, finite-width slab escape (Lemma 3) is proved using recurrent-site trials and first visits to new columns, in the spirit of (Zerner and Merkl 2001, Lemma 4). Second, scripts launched from a tilted record use pairwise distinct, previously unused rows. Their annealed cost is a product of positive row averages, rather than a product of uniform quenched lower bounds.

Suppose, toward a contradiction, that \(0<P_0(A_\ell)<1\). A slab-exit argument and fresh-half-space regeneration first show that the path escapes in one of the two signs almost surely. At a strict record from which the path never falls below its new height, the future factors from the past. The path between successive such records is a finite regeneration piece. Under conditioning never to backtrack, these pieces are independent and identically distributed. Their mean height width is finite, even for an irrational direction.

The first main step, Proposition 8, proves that coexistence of the two signs also forces finite mean spatial radius of a piece. A very large lateral excursion would create a record in a tilted direction. From that record, a fresh continuation escaping in the opposite sign has a probability bounded below independently of the excursion scale. This would give a fixed positive probability to arbitrarily large finite height maxima, contradicting the tail of their distribution. The radius bound gives deterministic limiting rays on the two escape events; the rays must be opposite. We may therefore reduce to coexistence in one coordinate direction.

We then sample two independent sequences of coordinate regeneration pieces. Place positive pieces downward by successively anchoring their terminal points, and place negative pieces downward in their ordinary chronological order. Individual path words retain their original orientation. Heights at which both sequences have a piece boundary divide the arrangement into independent common blocks. A contact is a site from which both arranged paths depart. For each dyadic interval of common-block indices, consider whether a contact occurs. Let \(m(J)\) be the sum of these probabilities over the first \(J\) such intervals, as defined precisely in (23).

The two final steps give \[m(J)=O\!\left(\frac{J}{\log J}\right),\qquad m(J)\ge c\,\frac{J}{\log\log J}\quad\text{for large }J,\] with \(c>0\), which are incompatible. For the upper bound, Proposition 14 compares the arrangement with independent bridges of prescribed heights. Randomizing the number of blocks keeps the cost of replacing each chunk by bridges of its realized height bounded. An independent stretch of blocks before the comparison interval turns the lateral alignment cost into an increment of displacement entropy. These increments telescope over dyadic scales, while finite mean widths make the bridge contact probabilities summable on those scales. A relative-entropy bound converts these two estimates into the stated bound for the expected number of contact intervals.

For the lower bound, view a positive bridge in chronological order and start the negative path just below its random endpoint. A union bound over all possible endpoints would be too expensive. Instead, the conditional probabilities of the possible endpoints form a vector of annealed martingales. Lemma 17 bounds the sum of their running maxima by an exponential tail on a logarithmic scale. In Proposition 18, first-contact prefixes are transferred to one shared environment before they are classified by quenched exit probabilities; the posterior functions remain in the unrevealed annealed path filtration. This separation controls the endpoint-dependent alignment without imposing a martingale property after the environment is revealed. A range of dyadic height intervals without a contact forces a late first contact near the top. Applying the estimate at the many regeneration boundaries in that range yields many contact height scales. The finite mean common-block width then converts the height count into the block-index count \(m(J)\).

Section 2 proves the path-weight and regeneration facts. Section 3 establishes the spatial moment bound. Section 4 reduces to coordinate heights and defines the common arrangement. Sections 5 and 6 prove the entropy and endpoint-posterior estimates. Section 7 gives the lower contact count and completes the theorem.

Departure rows and path weights

A finite nearest-neighbor word is a sequence \(\gamma=(x_0,\ldots,x_m)\) with \(x_{j+1}-x_j\in U\). Its departure set and annealed weight are \[\operatorname{Dep}(\gamma)=\{x_0,\ldots,x_{m-1}\},\qquad W(\gamma)=\mathbb E\!\left[ \prod_{j=0}^{m-1}\omega(x_j,x_{j+1}-x_j)\right],\] where the expectation in this display is over the environment. Thus \(W(\gamma)=P_{x_0}(X_j=x_j,\ 0\le j\le m)\). A row visited repeatedly contributes repeatedly to the same product inside the expectation. The terminal site contributes no row unless the word departed from it earlier. Translation invariance gives \(W(\gamma+z)=W(\gamma)\).

The transfer is made for unclassified path weights. A condition involving a quenched exit probability can depend on rows on both sides of the separation and cannot simply be inserted into the independent side of (2). Later we will first apply this identity and then classify sites in the shared environment.

Fresh records and true cuts

Fix \(v\in\mathbb R^d\setminus\{0\}\), normalized by \(\|v\|_\infty=1\), and write \(h(x)=x\cdot v\). Define \[D_v=\{h(X_n)\ge0\text{ for all }n\ge0\},\qquad p_v=P_0(D_v),\qquad \widehat P_v=P_0(\,\cdot\mid D_v)\] when \(p_v>0\). All statements in Section 2 concern this fixed real direction. We write \(\widehat E_v\) for expectation under \(\widehat P_v\). No arithmetic assumption on the coordinates of \(v\) is imposed.

The next lemma follows the repeated-trial approach of Zerner and Merkl (Zerner and Merkl 2001, Lemma 4). We give the argument to specify which transition rows remain unused at each trial.

The ordinary strict-record times are the stopping times \[\rho_0=0,\qquad \rho_{i+1}=\inf\{n>\rho_i:h(X_n)>h(X_{\rho_i})\},\] with all subsequent times set to infinity if one is infinite. A finite \(\rho_i\), \(i\ge1\), is a true cut if \[h(X_n)\ge h(X_{\rho_i})\quad\text{for every }n\ge\rho_i.\] Truth of a cut is not measurable at its arrival time. Whenever true cuts exist, \(\tau_1,\tau_2,\ldots\) enumerate them in time order.

At an ordinary strict record, every earlier departure has height strictly below the current height. The closed upper half-space is therefore unused. Applying Lemma 2 to each finite record prefix and translating its endpoint gives \[ P_0\big(h(X_{\rho_i+k})\ge h(X_{\rho_i})\text{ for all }k\ge0 \mid\mathcal F_{\rho_i}\big)=p_v \quad\text{on }\{\rho_i<\infty\}. \tag{3}\] This is an identity in the annealed path filtration. Conditional on the whole environment, the corresponding probability would instead be \(q_v(X_{\rho_i},\omega)\).

For a word \(\gamma=(0=x_0,\ldots,x_m=z)\), say that it is a regeneration word in direction \(v\) if \(m\ge1\) and the following two conditions hold:

1. \(0\le h(x_j)<h(z)\) for every \(0\le j<m\);

2. whenever \(1\le j<m\) is a strict record within the word, there is a \(k\) with \(j<k<m\) and \(h(x_k)<h(x_j)\).

The second condition says that every intermediate record has already failed before the final arrival. Define the width and radius of such a word by \[L(\gamma)=h(z)>0,\qquad R(\gamma)=\max_{0\le j\le m}\|x_j\|_2.\] Both are finite, but the width need not be an integer.

We call these relative words slabs. The preceding proof gives their complete law, not just independence of their displacements. This will allow us to arrange whole slabs in different spatial positions.

Mean width and the two signs

The record heights can have arbitrarily small positive increments in a real direction. The next argument instead renews in the integer index of an ordinary record.

Lemma 6 controls record count and projected width, not the number of time steps in a slab.

Thus a probability strictly between zero and one for \(A_v\) supplies positive probabilities for both signs. In Section 3, that coexistence assumption will give the stronger spatial estimate \(\mathbb E_{\nu_v}R+\mathbb E_{\nu_{-v}}R<\infty\).

Reduction to a coordinate direction

It now suffices to rule out a mixed coordinate direction. Relabel coordinates so that it is \(e_1\), and write \(p_\pm=p_{\pm e_1}>0\). Write \(D_\sigma=D_{\sigma e_1}\) and \(\widehat P_\sigma=\widehat P_{\sigma e_1}\) for either sign. The slab laws \(\nu_\pm=\nu_{\pm e_1}\) have positive integer widths; \(E_\pm\) will now denote expectation for these coordinate slab laws. Lemma 6 and Proposition 8, applied to this coordinate, give \[ E_\pm L<\infty,\qquad E_\pm R<\infty. \tag{16}\] All heights in the remaining proof are integer coordinate heights. In particular, a walk started inside a finite coordinate-height interval exits it almost surely: a path confined to that interval would have finite height supremum without tending to negative infinity, contrary to Lemma 3. Translation invariance and integration give this exit property quenched for almost every environment, simultaneously for all integer barriers and starting sites. Only a countable intersection is needed. This property does not require uniform ellipticity.

Exact-cut bridges and renewal tails

For a sign \(\sigma\in\{+,-\}\) define, with starting point zero, \[T_j^\sigma=\inf\{n\ge0:\sigma X_n\cdot e_1=j\},\qquad u_j^\sigma=P_0(T_j^\sigma<T_{-1}^\sigma),\qquad F_\sigma(j)=\nu_\sigma(L>j),\quad j\ge0.\] When only one sign is being discussed we omit \(\sigma\). A relative slab word starts at zero; its raw weight is the function \(W\) defined in Section 2.

Two tapes placed from a common top

Berger’s backward-path construction also assembles regeneration slabs from a terminal reference point (Berger 2008, secs. 2–3). In the arrangement below, placement proceeds downwards on both tapes, while each word retains its chronological orientation.

Take independent iid tapes \((A_i)_{i\ge1}\) and \((B_i)_{i\ge1}\) with respective word laws \(\nu_+\) and \(\nu_-\). Their joint law and expectation will be denoted by \(\mathbf P\) and \(\mathbf E\). For a relative word \(\gamma\) let \(d(\gamma)\) be its terminal displacement, and let \(a(\gamma)\in\mathbb Z^{d-1}\) be the last \(d-1\) coordinates of that displacement.

Set the positive top endpoint at zero. Place \(A_1\) with its terminal there, then place \(A_2\) with its terminal at the start of \(A_1\), and continue downwards. Thus the placed copy of \(A_i\) is translated by \(-\sum_{r\le i}d(A_r)\). Place the negative tape chronologically starting from \(-e_1\), so the placed copy of \(B_i\) is translated by \(-e_1+\sum_{r<i}d(B_r)\). Only translations are used; the steps inside each word retain their chronological orientation.

For a site \(x\), call \(-x_1\) its progress below the top, or depth. A word of width \(L\) preceded by total tape width \(s\) has all its departures in the progress interval \[ (s,s+L]\cap\mathbb Z \tag{20}\] on either tape. The one-level offset of the negative starting point is what makes these intervals agree. A contact is a site that is a departure of both placed tapes. The last positive step into the top is axial, so its departure is \(-e_1\), the first negative departure. There is always a contact at progress one.

Write \(H_i^+=\sum_{r\le i}L(A_r)\) and \(H_j^-=\sum_{r\le j}L(B_r)\), with \(H_0^+=H_0^-=0\). Successive positive common values of these two increasing sequences divide both tapes into common blocks. This is the intersection of two independent renewal processes, a standard construction discussed, for example, in (Alexander and Berger 2016). Lemma 13 proves that all such blocks exist and provides the moments needed for the entropy comparison.

Let \(K_j\) be the successive common widths and put \(W_0=0\), \(W_j=\sum_{i=1}^jK_i\). By (20), both tapes’ departures in block \(j\) have progress in \((W_{j-1},W_j]\). Consequently a contact always belongs to the same unique common block on both tapes. Figure 1 summarizes the two orientations.

For integers \(0\le a<b\), denote by \(\mathcal C(a,b]\) the event of a contact in at least one block with index \(a<j\le b\). The quantity to be estimated is \[ m(J)=\sum_{j=1}^J\mathbf P\bigl(\mathcal C(2^j,2^{j+1}]\bigr), \qquad J\ge1. \tag{23}\] It is the expected number of dyadic block-index intervals containing a contact. Section 5 bounds it above using the finite first moment of the lateral increment \(S\); the remaining sections obtain an incompatible lower bound from the forced contact near the top and the renewal-tail estimate.

A randomized comparison with independent bridges

Fix an integer \(n\ge1\). We want to cover the common blocks with indices in \((4n,8n]\) by a middle portion of the arrangement. We will surround it by two outer portions, each of height at least \(n\), so that any middle contact occurs well inside a finite slab. A buffer above all three portions supplies an independent lateral gap.

We randomize the number of blocks in each portion as well as in the buffer. Choosing a chunk count uniformly from \(n\) possibilities will offset the cost of specifying its random height. Randomizing the buffer count will let us compare its entropy with that of the whole experiment without paying an additional \(\log n\). Independently of the common-block tape, choose four independent integer counts \[ \begin{split} U,I_1,I_3&\ \hbox{uniform on }\{n,\ldots,2n-1\},\\ I_2&\ \hbox{uniform on }\{7n,\ldots,8n-1\}. \end{split} \tag{24}\] Read a buffer of \(U\) common blocks, followed by three chunks of \(I_1,I_2,I_3\) common blocks. Let \(d_0\) be the lateral gap after the buffer, namely the sum of its gap increments. For chunk \(i\), let \(A_i\) and \(B_i\) be the positive and negative slab lists in their downward tape order, and let \(H_i\) be their common total width. Write \[H=(H_1,H_2,H_3),\qquad \mathcal A=(A_1,A_2,A_3),\qquad \mathcal B=(B_1,B_2,B_3).\] The three chunks are independent, and \((U,d_0)\) is independent of all their data. To see this despite their random boundaries, condition on the four external counts and use the product law of the common blocks; the distribution of a chunk depends only on its own count. The first and third count windows ensure the required outer heights. The longer middle window ensures that the middle chunk starts before block \(4n\) and ends after block \(8n\).

Set \(q=d-1\), and let \(X_{A,i},X_{B,i}\in\mathbb Z^q\) be the sums of the lateral displacements of the words in \(A_i,B_i\), measured in the words’ actual chronological directions. The total positive displacement determines where its bottom lies relative to its top. Accordingly, define \[ X_A=\sum_{i=1}^3X_{A,i},\quad X_B=\sum_{i=1}^3X_{B,i},\quad Z=d_0+X_A,\quad T=U+I_1+I_2+I_3. \tag{25}\] Here is the placement that explains the alignment variable \(Z\) and will define the same contact event under the actual and reference laws. Set \(N_*=H_1+H_2+H_3\). Place the positive chunks from height \(0\) and lateral position \(0\), in chronological order \(3,2,1\), reversing the list order within each chunk. Each word is still traversed in its original direction. Place the negative chunks in chronological order \(1,2,3\) from height \(N_*-1\) and lateral position \(Z\). Let \(E_{\mathrm{mid}}\) be a departure contact between the two middle chunks.

For these actual chunk data, this placement reproduces the infinite opposed arrangement after its buffer, up to translation. For example, at the top of the middle chunk the lateral gap, negative minus positive, is \[Z+X_{B,1}-(X_{A,3}+X_{A,2}) =d_0+X_{A,1}+X_{B,1},\] as in the original arrangement; the subsequent list placements also agree. From (24), the middle chunk starts after at most \(4n-2\) common blocks and ends after at least \(9n\) blocks. It therefore contains the deterministic common-block interval \((4n,8n]\).

Denote the actual law of \((H,Z,\mathcal A,\mathcal B)\) by \(P_n\). Define a reference law \(Q_n\) as follows. First sample \((H,Z)\) with its actual joint marginal. Given these values, sample the six lists independently, with respective exact-cut bridge laws \(\mathcal B_{H_i}^+\) and \(\mathcal B_{H_i}^-\) from Lemma 11. The reference retains the dependence between \(H\) and \(Z\); it changes only the conditional law of the lists. Apply the same placement rule to its newly sampled lists. Thus \(E_{\mathrm{mid}}\) is one fixed measurable event in both experiments, and its actual probability bounds the original contact probability in \((4n,8n]\).

The entropy cost of the comparison

We use entropy and relative entropy in their classical information-theoretic sense (Shannon 1948; Kullback and Leibler 1951), recalling the identities and bounds needed for the countable path spaces below.

For a countable random variable \(V\), write \[\mathcal H(V)=-\sum_v P(V=v)\log P(V=v),\] with \(0\log0=0\). Conditional entropy is the average of the corresponding conditional entropies. For probability measures \(P,Q\) on a countable set, their relative entropy is \[D(P\Vert Q)=\sum_x P(x)\log\frac{P(x)}{Q(x)},\] with value \(+\infty\) if \(P\) is not absolutely continuous with respect to \(Q\). Conditional mutual information \(I(V;W\mid H)\) is the average, over \(H\), of the relative entropy between the joint conditional law and the product of its two marginals.

We use the chain rule for relative entropy, obtained by factoring joint probabilities into marginal and conditional probabilities. In particular, if a reference law for \((V,W)\) conditional on \(H\) factors between \(V\) and \(W\), the averaged conditional relative entropy is at least \(I(V;W\mid H)\). Nonnegativity of relative entropy follows from Jensen’s inequality applied to \(-\log\). The same factorization, followed by nonnegativity, shows that applying a measurable map cannot increase relative entropy. These facts apply to countable path lists without assuming that the lists themselves have finite entropy; equivalently one may first use finite partitions and pass to their increasing limit.

Let \(S_1,S_2,\ldots\) be the iid lateral gap increments of the common blocks, and set \[h_k=\mathcal H(S_1+\cdots+S_k),\qquad k\ge1.\] Lemma 13 gives \(\mathbf E|S_1|<\infty\). For a \(\mathbb Z^q\)-valued variable with finite first moment, comparison with the probability weights proportional to \(\exp(-|x|/(k+1))\) gives \[ h_k\le C+q\log(k+1). \tag{26}\] Indeed the normalizing sum is at most \(C_q(k+1)^q\), and the expected comparison energy is at most \(k\mathbf E|S_1|/(k+1)\). Also \(h_k\) is nondecreasing: condition on the last independent summand, use translation invariance of entropy, and then remove the conditioning.

By Lemma 13, the sum of the slab radii in a common block has finite expectation. Thus each tape’s block displacement, and not only their sum \(S\), has a finite first absolute moment. In addition the common width has finite mean. All count, height, and displacement variables used below therefore have first moments of order \(n\), and their joint and conditional entropies are finite. No entropy bound for the full path lists is needed.

Rare contacts under the reference law

Comparing opposed paths in one environment and classifying intersection sites by quenched escape probabilities already appears in the planar argument of Zerner (Zerner 2007, sec. 2). We prove the finite-barrier estimate needed here.

Summing over scales

The estimate uses the same departure contacts as the original arrangement. The next section bounds the chance of a first contact far above a true cut; that bound will force more contact scales than Proposition 14 permits.

The sum of posterior maxima

The first-moment calculation integrates the nonnegative-martingale maximal inequality, whose classical form appears in (Ville 1939, V, Première section, §2, Theorem 1). The same coordinatewise calculation for posterior vectors appears in Kesselheim, Molinaro, Patton, and Singla (Kesselheim et al. 2026, sec. 1.1). To obtain an exponential moment, we also control conditional future increases and the size of each jump. This is the increasing-process energy mechanism of (Kikuchi 1992, Theorem 4 and its corollary); we give an elementary threshold proof with the required filtration explicit.

A first-contact estimate

The comparison of opposing paths through unvisited rows and the use of quenched exit probabilities are related to Zerner’s planar argument (Zerner 2007, sec. 2). Here the negative path starts at a random endpoint of the positive bridge; the posterior estimate controls the dependence created by that alignment.

We continue to use integer height \(x_1\), and write \(\pi x\in\mathbb Z^{d-1}\) for the lateral coordinates of \(x\in\mathbb Z^d\). Recall that \(u_n^+=P_0(T_n<T_{-1})\) and \(p_\pm>0\) are the non-backtracking probabilities. The constants below may depend on the environment law and the dimension. We assume only strict positivity of the transition rows; no common ellipticity bound is imposed.

Fix an integer \(N\ge3\). Let \(Y\) have the annealed bridge law from \(0\) to its first arrival at height \(N\), conditioned on \(T_N<T_{-1}\), and put \(E=\pi Y_{T_N}\). Independently sample a negative path \(B\) with law \(\widehat P_{-e_1}\). Translate it to the random starting site \((N-1,E)\): \[\beta_j=(N-1,E)+B_j,\qquad j\ge0.\] Thus the relative negative path is independent of the positive bridge; its absolute position depends on the bridge endpoint. A departure of \(Y\) means an index \(i<T_N\). Every site of the infinite path \(\beta\) is a departure site. For a negative path started at height \(N-1\), write \(D^-\) for the event that it never rises above \(N-1\).

The estimate is useful when the remaining gap \(N-r\) is small relative to the height \(r-k\) already crossed. Lemma 12 gives \(\Delta\le (N-r)F_+(r-k)\); Section 7 arranges precisely this separation of scales.

The time \(\tau\) refers to the entire negative trajectory and is not asserted to be a stopping time for the positive path. We will enumerate finite prefixes to use its first-contact property. Martingale estimates will instead be stopped at deterministic height barriers.

The estimate controls a contact anywhere after the prescribed height \(r\). It therefore applies to a long interval without subdividing the positive path by its successive record heights. In the next section a missing interval of contact depths forces exactly such a late first contact.