A Stationary, Mixing and Perturbative Counterexample to the 0-1-law for Random Walk in Random Environment in Two Dimensions

We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non-trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d.\ environments.


Random walk in random environment
We start by fixing the notation and the basic notions of the model. We will count dimensions from 0 to d − 1; so, we write u = (u 0 , u 1 , . . . , u d−1 ) ∈ Z d , and denote by e 0 , . . . e d−1 the canonical unit vectors in Z d . This nonstandard-notation will simplify things later. For two vectors v, w ∈ Z d , v · w denotes the scalar product.
For any real number r ∈ R, we will be using the floor function r := max{m ∈ N 0 : m ≤ r} and for any natural number l ∈ N 0 the modulo operation l mod 2 := 1 l is impair ∈ {0, 1}.
If P is a probability measure, with the convenient notational abuse common in mathematical physics, we write "P" for the expectation operator as well. Define S d := ∈ [0, 1] {±ej ,0≤j<d} : e∈{±ej ,0≤j<d} (e) = 1 , d ∈ N, the set of nearest neighbour transition probabilities on Z d . We call a family ω = (ω u ) u∈Z d of S d -valued random variables on an appropriate probability space (Ω, A, P ) a random environment on Z d .
(1.1) If (1.1) is satisfied with κ = 0, the environment is called elliptic, and if it is even satisfied with some κ > 0, uniformly elliptic.
We use the term totally ergodic for "ergodic with respect to any shift".
Take a starting point v ∈ Z d . To a random environment ω on (Ω, A, P ), we associate the random probability measure P ω v , which, together with the Z d -valued random variables (X t ) t∈N0 , establishes the random walk in random environment (P, P ω v , (X t ) t∈N0 ). It is defined to satisfy the Markov-property and In [Kal81], Kalikow considered questions of recurrence and transience of this model, and proved that for uniformly elliptic i.i.d.-environments, P P ω 0 (X t · v changes sign infinitely often) ∈ {0, 1}, v ∈ Z d . (1.3) He also raised the question whether in d = 2, it holds that (1.4) Sznitman and Zerner highlighted in [SZ99] that Kalikow's question (1.4) is valid in any dimension d ≥ 2. They also pointed out that (1.3) implies The term Kalikow's 0-1-law has since been established for this assertion. For d = 2, Zerner and Merkl answer Kalikow's question (positively) for elliptic i.i.d.-environments in [ZM01]; an improved version of the proof is given in [Zer07]. Holmes and Salisbury treat the same questions without the assumption of ellipticity in [HS].
The necessity of the i.i.d.
-assumption is assessed in [ZM01] by means of an example for d = 2 of an elliptic, ergodic and stationary environment that features (1.5) [Zer07] gives a similar example with an even totally ergodic environment. As for d ≥ 3, Bramson, Zeitouni and Zerner [BZZ06] have a uniformly elliptic, stationary, totally ergodic, and even mixing example of an environment satisfying (1.5).
In the present article, we construct an environment with similar properties for dimension d = 2. Our main theorem is indeed: Theorem 1.0.1. For any ε > 0, there is an ε-perturbative, stationary, mixing random environment ω = (ω u ) u∈Z 2 with associated probability measure P such that for the associated random walk ((X t ), P ω 0 ), it holds that The ideas of conducting the random walk to infinity on a "treelike structure" of "not too slowly growing roads leading to infinity" has been applied in [BZZ06]. As for how to construct such a structure in dimension d = 2, Häggström and Mester [HM09] had the idea of ever larger, ever rarer streets joining each other. By using Poisson processes of different intensities as the underlying structure instead of their"windows" of fixed length, we were able to avoid some of the rigidity of their model and to make assertions on mixing, at the price of developing a completely new construction.
2 Construction of a random environment 2.1 Notation
For v, w ∈ Z 2 we define the box between v and w as The (outer) boundary of a box B may be defined as here, d(·, ·) means the 1-metric. It is convenient to define as well the closure of B, which is iii the upper right corner B of a box B is

Streets and streetgrid, and blocks
We call a number m ∈ N 0 a superlevel, and k ∈ {0, 1} a sublevel. The mapping (m, k) → 2m+k : N 0 × {0, 1} → N 0 is bijective, and this number is called the corresponding level. Given any level l ∈ N 0 , we can obviously reconstitute superlevel and sublevel using the inverse function, ( l 2 , l mod 2).
If a level has somehow been assigned to some object, we will speak of the superlevel and the sublevel of that object as well.
Given a level l ∈ N 0 and a function F We call it a field w.r.t. F if it is a street of level 0 w.r.t. F . When it is obvious or not important which level and function are meant, we will simply speak of "street" and "field".
We say F is a streetgrid if D is the union of streets and fields with respect to F , i.e.
Given a box B contained in the domain of a streetgrid F , we define the level of the box B w.r.t. F as Note that if the box B is a street, the two definitions of "level of the box B" and "level of the street B" coincide.  iv are called the rate of occurrence of streets at superlevel m and the planned widths of the streets at superlevel m, respectively. We define and, on some appropriate probability space (Ω, F, P ), a family of independent random variables which are to be Bernoulli-distributed with parameters λ l 2 . To understand the meaning of the index-set Z, we need to read it backwards. Every point in Z 2 gets for every level in N 0 a Bernoulli-process {0, 1} Z .
Having the necessary terms and definitions as well as the random ingredients at hand, we can start constructing the environment, beginning with a streetgrid. This will be done in two steps. Starting at the origin, we begin with narrow streets and make our way towards infinity by ever wider ones. This leaves wide areas of fields that will then be filled in the opposite direction with ever narrowing streets.

The initial grid
We could put the random ingredient X directly into our construction, which will be built gradually in several definitions. We prefer however to write down these definitions as functions on {0, 1} Z , and to finally evaluate them at the random place X. Notationwise, we will drop the dependence on x ∈ {0, 1} Z after the first appearence, though. Please note that the definitions may, for some x ∈ {0, 1} Z , not make any sense; whenever there is some doubt on how x should look like, any typical realization x of X will do.
In a first step, we define processes that, roughly speaking, show where streets would be if each coordinate existed on its own. For each coordinate direction j ∈ {0, 1} and every superlevel m ∈ N 0 , we attach to the left of every point highlighted as 1 by the process x(·, 2m + j, 0) an interval with the width β m of the respective superlevel m. Then, for any point, we take the maximum level of all streets the point lies in; that is, in the case of overlapping intervals of different levels, the higher level prevails: W j x (x) := 2 max m ∈ N 0 : ∃y ∈ Z : x ≤ y < x + β m , x(y, 2m + j, 0) = 1 + j, We need to make sure W j x (X) is P -a.s. finite for all j ∈ {0, 1} and all x ∈ Z. For m ∈ N, x ∈ Z, it holds that With the Borel-Cantelli-lemma, we conclude that there are P -a.s. only finitely many m ∈ N satisfying the condition of the maximum in (2.5), which hence is P -a.s. finite.
The dependence on x will de dropped for the next few definitions, even though it of course persists. The function W j x will be further transformed by removing the outer intervals of smaller value in Note that the maximum over an empty set is to be read as −∞.
The transition from W 0 · to V 0 · is visualized in Figure 1.
1} are not monotonically increasing, but "weakly monotonically increasing, seen from 0" in the sense that they still satisfy and a similar assertion for negative x.
With the following definition, we begin our two-dimensional construction. Any point u = (u 0 , u 1 ) ∈ Z 2 gets assigned a level by In words, the point u gets assigned the maximum of the two V 0 u0 and V 1 u1 provided this maximum is larger than any of the V j x for x between 0 and u j , with j ∈ {0, 1}. Thus, InitGrid satisfies a two-dimensional analogue of the heuristical notion of "weakly monotonically increasing seen from 0" mentioned in Remark 2.2.1.
Proof. We need to show that Z 2 is a patchwork of streets and fields w.r.t. InitGrid X (·) as in (2.3). We will concentrate on the first quadrant, referring to analogy for the other ones.
On the coordinate axes, we have that On the first quadrant, it holds that , and more generally, If one takes this equation for fixed, say, x with V 0 x = 0 and lets run y from 0 to infinity, one gets the value InitGrid((x, y)) = V 0 x = InitGrid(xe 0 ) for all y < min{y ∈ N 0 |V 1 y > V 0 x }; in other words, until from the other coordinate, one gets blocked. Because V 0 · and V 1 · have disjoint codomains (except for 0, which they have in common), these blockings are sharp in the sense that one can always tell whether a point has got its value (different from 0) from V 0 · or V 1 · .
Also, the other way around, if some point (x, y) ∈ Z 2 has got its initial-grid-value from, say, V 0 x , then fixing x and letting z run from y to 0 yields InitGrid (x, z) = InitGrid (x, y) for all y ≥ z ≥ 0.
Combining the arguments of the last two paragraphs, one can see that all points u ∈ Z 2 satisfying InitGrid(u) = 2m + j = 0 lie in areas of constant InitGrid-value outgoing perpendicularily from the j-th coordinate axis. Each such area continues until it gets blocked by some area coming from the other coordinate axis. The areas are of rectangular shape, and P -a.s. finite. This applies as well to the areas where the initial grid equals 0. These are indeed surrounded by four streets of different levels, so that they are fields.
Finally, we need not only to pay attention at the the four quadrants individually, but at the transition between them as well. Indeed, the streetgrid-property holds because between adjacent quadrants, the same V j · , j ∈ {0, 1} influences the construction of the streets.
Remark 2.2.3. We will be saying " 0 is responsible in InitGrid for the emplacement of streets of level l on D" for any block D w.r.t. InitGrid containing the origin and any level InitGrid (D) ≤ l < InitGrid (D).
InitGrid (D) is the highest level of any streets placed on D. The emplacement of these streets has been provided by the random ingredient X evaluated at points (·, l, 0), so it is sound to say 0 is responsible.
How about the levels InitGrid (D) < l < InitGrid (D)? No street of these levels exists in D. But this absence of streets was stipulated by an absence of 1s in the random ingredient X at points (·, l, 0), InitGrid (D) < l < InitGrid (D). So it is legitimate to say 0 is responsible for those levels as well.
We will extend the notion of responsibility in Definition 2.2.5.

Asphalting of the remaining fields
After constructing InitGrid x (u), we continue by iteratively putting the missing streets on the remaining fields. Let us describe informally how we proceed. The streets that are not fields w.r.t. InitGrid x are to remain untouched. We want to work exclusively on the fields.
By Lemma 2.2.2, any field B w.r.t. InitGrid x is surrounded by four streets. The minimum of their level minus one is the level of the first streets that should be put on B. Determining the level of the streets to put is hence the first step.
Then, we need to know the place where we put these streets. To each field B will be assigned an own process resembling the one in (2.5); this time however, only one level at a time is taken into account. The random ingredient of this process will be the Bernoulli process associated to the upper right corner of B and the respective level. Now, when the streets are put on the fields, smaller fields are created; on these, we put streets of the next lower level, and so on. Now, back to rigid definitions. First, we define a dummy and the starting point of the iteration, For i ≥ 1, and B a field with respect to the i-th iteration step L i · , we associate a level to B by This is the level of the streets that are going to be placed on B. The first line of the definition is used at the first iteration step, and also the default for the following steps; only if there has no street been put on a field in the last step, the second line makes sure that in the current step, the same level is not used again. We provide the emplacement in B for the new streets of level l (we exceptionally remind the dependence on x) by Given l, the indicator function checks whether at the point x, there is a street of level l induced by the Bernoulli process at the upper right corner of the field. The streets are placed on the field B using The sublevel l mod 2 is taking care of the (vertical or horizontal) orientation of the streets.
ix We need to do this setting of streets in every field, and set the whole iteration step as We have put, on every field w.r.t. L i−1 · , streets of "one level lower". The process L i · = L i · (x) converges pointwise with i → ∞ for P -almost any realization x of X: for any field w.r.t. InitGrid x (·), at some iteration, the level 1 (with superlevel 0) is reached and the remaining sub-fields are entirely filled with streets of level 1. Another way of seeing the convergence is by remarking that for every point u ∈ Z 2 , the sequence (L i u ) i∈N is monotonically increasing and bounded. The limes will be called the final streetgrid SG(x) = (SG(x) u ) u∈Z 2 and we write SG = (SG u ) u∈Z 2 := (SG(X) u ) u∈Z 2 .
Based on the earlier simulation of the initial grid, a simulation of the final streetgrid can be found in Figure 3.
Proof. Each iteration step L i is: to obtain L i , only the fields of L i−1 are changed, and on these fields are placed streets extending in one coordinate direction up to the boundary of the field they are placed on. These streets are of strictly lower level than all surrounding streets.
Because the passage to the limit is of the type where for any finite region, the sequence is from some point on constant and equal to the limiting object, SG is a streetgrid as well.
We turn again towards the concept of responsibility. This time, we give a precise definition, and then explain how it relates to our construction of the streetgrid.
Definition 2.2.5. Take a streetgrid g. For any block D w.r.t. g and any g (D) there is a unique w ∈ Z 2 of which we say that it is in g responsible for the emplacement of streets of level l in D. It is given by w = 0 if 0 ∈ D, and w = D if 0 ∈ D.
For g = InitGrid, this definition exactly reflects Remark 2.2.3. The streets already present in InitGrid are carried over to SG, so it is reasonable to say 0 is responsible for these in SG as well.
The responsibility of points w = 0 can be understood as follows: Any field D w.r.t. InitGrid does not contain the origin. It is also a block and will remain a block in the course of the construction.
The first iteration step is about placing streets of level l = InitGrid (D) − 1 on D. The randomness for their emplacement comes from the Bernoulli process X((·, l, D)). This is why D should be considered responsible for this block and level.
If no streets of level l are placed (because the Bernoulli process is 0 in the relevant range), D is responsible for the subsequent lower levels as well, until streets is placed. The level of these streets will later turn out to be the level SG (D) of the block D.
By the placement of these streets, smaller fields are created, and it is their upper right corner that provides the randomness via X. These upper right corners are hence the places that are responsible for the streets of these lower levels, on these smaller fields (which again are and remain blocks).
The next level shows that there is no conflict of responsibility.
Lemma 2.2.6. Take a streetgrid g. If w ∈ Z 2 is responsible in g for the emplacement of streets of level l in D, where l ∈ N is a level and D some block w.r.t. g, then w is not responsible in g for the emplacement of streets of level l in B, where B = D is some other block w.r.t. g.
x Figure 3: Simulation of the final street grid SG. Where was the origin again?
xi Proof. Suppose w is responsible in g for the emplacement of streets of level l in both D and B, where both D and B are blocks w.r.t. g, but D = B. A first deduction is that either both B and D must contain the origin, or share the same upper right corner w = B = D. In either case, B ∩ D = ∅. As B = D, this implies that, without loss of generality, . This is a contradiction to that w was to be responsible for the same level in B and D.

Transition probabilities for the random environment
In order to determine where what transition kernels will be placed, we cut down the streets of the streetgrid to lanes using the following definition: ≥ 2 and sublevel k := SG (B) mod 2, we define the lanes The definition of b · (B) and b · (B) was given in (2.2).
Note that there might be some non-empty space between the two middle lanes Lane SG +,− (B) and Lane SG −,− (B). We want to place the transition probabilities in a way that on the lanes with "+" as first index, the random walk feels a drift northwards or eastwards (if the sublevel of the street is 0 or 1, respectively), and on the lanes with "−" as first index, it feels a drift to the south or the west. The distinction between + and − in the second index is then used to provide a drift to the area where two lanes of the same street with the same first index meet.
We will also be using the notation ω = ω +,+ and visualize this local transition probability either by or . See also Figure 4. We will need a reflection matrix, namely to place the transition probabilities we just defined on the streets.
xii Figure 4: The transition probability kernels ω j ·,· . The lengths of the arrows are not to scale.
Definition 2.3.3. Given the streetgrid SG(X), the transition probability kernels of the environment at place u ∈ Z 2 will be defined as follows. If u ∈ B a street w.r.t. SG(X) such that If u ∈ B any other street, set ω u := ω 1 4 .
A visualization of the lanes and the different corresponding transition probabilities can be found in Figure 5. for all u ∈ Z 2 such that u 1 ≥ 0, ω for all u ∈ Z 2 such that u 1 < 0.
That is, the random walk is subject to a uniform drift in direction of e 0 and towards the zeroth coordinate axis. It is easy to prove by standard martingale methods and the Borel-Cantelli-Lemma that the associated random walk in random environment (X n ) n starting at 0 has positive probability never to leave the set {x ∈ Z 2 |x 0 ≥ 0, |x 1 | ≤ √ x 0 }, while following the first coordinate axis to infinity. The morality of this example is that a random walk with uniform drift along a line and with a drift pushing it back towards that line has positive probability to never be further away from the line than the square root of the travelled length.
We will prove that P -almost surely, somewhere, there is a street w.r.t. InitGrid X on the first coordinate axis satisfying the following: if one walks down that street (northwards) until one hits a perpendicular street, walks eastwards on that new one until the next perpendicular street, starts walking northwards again, and so on; if one does so, then: • at the end of one street, one always encounters one of the next higher level; • the width of these streets grows nicely, • the streets are not too long.
Also, there will always be a drift pushing forward and to the middle of the two lanes with first index "+" in these streets.
The idea is that, when walking like described above, the width of the street the walker is in as a function of the distance travelled is larger than the square root (·) 1/2 ; this is in analogy to the above example.
An average-case-analysis shows heuristically why this is the case. The streets of superlevel m have a planned width of β m and, on average, a length of less than 1 λm+1 . The somewhat worst case for the random walk is if it has to go through the whole length of every street. The width of the n-th street the random walk visits is β n = n! 2 . The distance travelled is of the order of This shows that the square root of the travelled distance is of slower growth than the width of the streets, leaving enough room to the random walk for fluctuations without leaving the sequence of streets. The exact proof stretches over the whole subsection, but the most pertinent statements can be found in Corollarys 3.2.3, 3.2.7, and 3.2.8.
Remark 3.1.1. The statements above are even true with any root (·) 1/α , α > 1, instead of the square root (·) 1/2 . Hence we need to fix the exponent. Set α > 1 for the rest of the article. xiv

The way to infinity is eventually large
We start the proof of the above claims with a seemingly technical Definition and Lemma.
, are geometrically distributed, independent random variables with success probability λ m = 1 (m+1)! 2 . We can calculate, for the first event, We see that the first term converges, while the second one is summable, so that we can conclude using the Borel-Cantelli-Lemma. The probability of the second event computes just the same way, only the limit of the leading term is some other constant.
For the last event, we observe that is indeed summable as well.
Corollary 3.2.3. The event holds P -a.s. eventually. Hence, P -a.s., (3.2) M = M (X) is the superlevel from which the event defined in (3.1) always holds. The restriction to m ≥ 5 is made so that we do not have to worry about whether we can divide streets into four lanes, and subdivide lanes in four equal parts: already β 4 = 576 = 36 * 16.
There is a picture relating the terms of the event (3.1) in Figure 6. Proof. Take any m ≥ M . We have X(G 1 m , 2m + 1, 0) = 1 induces a (part of a) street of superlevel m, with planned width β m . Thus, this street of sublevel 1 does not reach the zeroth axis. Now take m > m. We know that G 0 m ≥ G 0 m −1 + β m ≥ G 0 m + β m . This shows that any vertical street of higher level does not reach G 0 m e 0 .
Proof. We prove the case j = 0. Take m ≥ M . As G 0 m is a natural number such that X G 0 m , 2m, 0 = 1, we have W 0 The case ">" subdivides into the two • InitGrid X (G 0 m e 0 ) = 2m + 1 for some m ≥ m, • InitGrid X (G 0 m e 0 ) = 2m for some m > m, which both are excluded by Lemma 3.2.4. Hence, equality holds in (3.3). G j m depends only on X(·, ·, 0). As all streets that are not fields w.r.t. InitGrid remain untouched in the construction of the final streetgrid, the equality holds for SG G 0 m e0 as well.
Proof. Both assertions follow from the same type of arguments as in the proof of the Lemmas 3.2.4 and 3.2.5; the second one makes also use of the upper bounds provided by (3.1). Proof. We prove only the first assertion. Let m ≥ M . As we have seen in Lemma 3.2.5, (B 0 m ) = 2m. By definition, the street B 0 m extends vertically until it is blocked by some horizontal higherlevel-street. The superlevel of this street is greater as or equal to m, otherwise there would be no blocking. Any horizontal street B 1 m of level m > m does not interfere with B 1 m , because the G 1 · all keep their distance from each other (see (3.1)). So, the blocking indeed happens by B 1 m . we shall express that for all x ∈ {0, 1} Z such that x| I =x, it holds that F (x) u = g u for all u ∈ D.

Stationarity
xvii Here, x| I : I → {0, 1} denotes the usual restriction of the function x : Z → {0, 1} on I. Notation (3.4) however is more restrictive than a mere restriction, because it is understood that on D, F (·) does not depend on the values at places in Z \ I. Also define 0| I to be the constant mapping that assigns 0 to any element in I. Proof. Take B a box containing the origin. For j ∈ {0, 1}, define the random variables These are P -almost surely finite, and B ⊆ D := {d 0 , . . . , d 0 } × {d 1 , . . . , d 1 }. D is a random set and a block w.r.t. InitGrid(·). The streets placed on D by the iterative construction leading to SG are all of lower level than the minimum of the levels present in ∂D, so that the block-property is preserved.
Definition 3.3.3. Let D be a block w.r.t. g ∈ N 0 D such that 0 ∈ D. Note that this is more a condition on g than on D. Define The dependence of I g and J g on D is omitted because it can be considered implicit via g.
To explain the meaning of these two sets, we need to go into greater detail. Take a realization of X. It is an element of {0, 1} Z , and leads to SG = SG(X). One can ask at which points in Z the values of X may be changed without changing the outcome of SG, or SG | D for some fixed D ⊆ Z.
The other way around, given a certain realization g ∈ N 0 D of SG(X)| D , where D ⊆ Z 2 is a box, one can ask about the set of realizations of X such that SG(X)| D = g.
It turns out it is enough to look at the outcome of X on the two subsets I g and J g of Z in order to decide whether the last equation is true or not. All points that are responsible in the sense of Definition 2.2.5 are contained in I g , and X(·) being equal to 0 at all points in J g stipulates the absence of big streets that are not supposed to be on D.
Let us look at the lines one at a time.
As D is a block w.r.t. g, and 0 ∈ D, all four streets in ∂D are already present in InitGrid x . InitGrid x is only influenced by the values of x at points (·, ·, 0). The streets w.r.t. g in D are either streets w.r.t. InitGrid x or are influenced by the values of x at the upper right corners of fields w.r.t. InitGrid x or the subsequent iteration steps in the construction. These fields are entirely contained in D, again because D is a block w.r.t. g. This is why points (·, ·, u) with u ∈ D have no influence.
We just looked at the influence of points in the upper right corners of fields lying entirely in D. The streets they induce are all of lower level than the minimum level present in ∂D; higher levels are not even considered, and thus the values of x at the points in the second line have no influence on the equation.
The values of points with lower level do have an influence, but only if the index of the Bernoulli-process is not too far from D; to be precice, neither left to the lower end in the j-th coordinate-direction of D, nor farther than one street-width to the right of the upper end of D.
Similarily, the values at the origin do not have any influence if the index of the Bernoulliprocess is too far from D; this translates as slightly loosened boundaries in the last line.
All remaining points are contained in I g and J g , which contain however some of the cases above as well. This proves that SG(x)| D = g does not depend on x| Z\(Ig∪Jg) . (3.6) can be written as x| Jg ≡ 0, which can hence be seen as an equivalent to (3.5). Finally, we have, with some non-trivial, non-random constant c, This value to be larger than zero is equivalent to and we can, by the finiteness of the sum, confirm positive P X| Jg -measure for 0| Jg . xix Definition 3.3.5. We need to define some shift operators and related notations. Let v ∈ Z 2 be the vector we want to shift by.
A slightly different shift will sometimes be needed for elements of the form (x, l, 0) ∈ Z, namely one that preserves the special role of the origin: With these last two definitions at hand, we can shift the two I g and J g from Definition 3.3.3 in the standard way by Finally, we shift whole configurationsx ∈ {0, 1} I , I ⊆ Z by defining Lemma 3.3.6. Let D 0 be a block w.r.t. g ∈ N 0 D , I g , J g from Definition 3.3.3. Also take any v such that −v ∈ D. Then, I θvg = θ v I g and J θvg = ϑ v J g , and for any x ∈ {0, 1} Z , SG(x)| D+v = θ v g implies SG(x| θvIg∪ϑvJg )| D+v = θ v g.
Proof. The first two equalities are easy exercises; an important point is how ϑ · preserves the special role of the origin, but at a different position relative to the shifted box.
The second assertion then follows directly frome Lemma 3.3.4, which tells us that SG(x)| D+v = θ v g implies SG(x| I θv g ∪J θv g )| D+v = θ v g. Figure 7 gives an idea of how the responsibility changes when the point of reference (the origin) is changed. This sort of changing will be employed in Definition 3.3.7 in order to create a configuration of {0, 1} Ig that yields the same outcome of the final streetgrid's construction, only shifted.
Definition 3.3.7. Take the hypotheses of Lemma 3.3.6. We define yet another operator on configurations on {0, 1} Ig , So, we need to define the object x (·) for all (y, l, w) ∈ I θvg . We do this first for a special case of pairs (l, w), and then for the rest.
Take any block B w.r.t. g, and g (B) ≤ l < g (B). Recall that B + v is a block w.r.t. θ v g, and that g (B) = θvg (B + v) and g (B) = θvg (B + v). So, we can apply Definition 2.2.5 and obtain w ∈ D and w ∈ D + v such that For any other case that has not yet been covered, take l < g (D) and w ∈ D + v such that • w is not in θ v g responsible for the emplacement of the streets of level l in B for any block B w.r.t. θ v g, and • w − v is not in g responsible for the emplacement of the streets of level l in B for any block B w.r.t. g; then, with m := l 2 , j := l mod 2, we define, This last definition shows that the operator is for most of the points really just the shift operator applied to the functionx; only at the few points that are responsible, and at their counterparts in the shifted set, the special definition takes effect, and so to say, the responsibility is switched.
Note that depends strongly on v and g, which gives again an implicit dependence on D. It also depends on our choice of I g and J g .
is well-defined, and takes indeed values in the specified codomain. It is bijective, and probability-preserving in the sense that Also, the following equivalence holds: Proof. For the first part of the definition, we remark that if w = w + v, the two definitions in (3.7) coincide: w = w − v. So, the two do not contradict each other immediately. Also, given any l, w and w are, respectively, responsible for l only in B and B + v. This was shown in Lemma 2.2.6. In the second part, the two bullets make sure that only cases not yet covered by the first part are defined. So, we indeed did not commit the error of multiply defining things.
The verification of x ∈ {0, 1} θvIg consists in checking that the domain of x is contained in θ v I g . Indeed, the indices y are chosen in the correct range. Also, l < g (B) < g (D). Finally, w, w + v ∈ B + v ⊆ D + v. The same applies to the second part of the definition.
To prove the bijectivity of = (v, g), we consider the inverse function, which is (−v, θ v g). To check that this is true, remark that the two parts of the definition of can be inverted separately; the responsible points are just reversed, and the responsibilities switched back. The points which are not responsible being identical, the values there get shifted back as well.
For the preservation of probability, note that leaves the levels intact, and replicates the same number of zeros and ones, just at different places. Then, the stationarity of the Bernoulliprocesses takes effect.
The last statement is a consequence of the concept of switching responsibilities described above. The operator moves the values ofx at any point responsible in g for the emplacement of streets of level l in B to the point which is in θ v g responsible for the emplacement of streets of level l on B + v. If one translates the concept of responsibility into the construcion of the streetgrid, one sees that SG( , 0| J θv g ) reconstitutes indeed the shifted g on the shifted domain.
The opposite inclusion follows from the above considerations on bijectivity.
Proof. We need to show the invariance of SG(X)'s finite-dimensional marginal distributions under the arbitrary shifts in Z 2 . Fortunately, we can restrict ourselves to distributions on boxes and shift-vectors inside these boxes: if we need a farther shift, we just take a bigger box.
Let B 0 be a box, v ∈ Z 2 such that −v ∈ B, and g ∈ N 0 B .
In the following calculations, the first equality is due to Lemma 3.3.2, the second one is true because the smallest (w.r.t. the semi-order established by the subset-relation) block around B is unique. The fourth equality holds because the block property depends only on D, and for the sixth one we apply Lemma 3.3.4 for one inclusion, the other one following directly from Notation 3.3.1. Lemma 3.3.4 also implies the disjointness of I g and J g leading to the independence used for the seventh equality. For the last equality, we apply Lemma 3.3.8.
We continue by applying the bijectivity of , and reverting the steps which lead here, but with respect to the shifted sets.
At this point, we need to adjust the summation. We perform some trivial shift operations and change the indices of the sums:  On the other hand, because −v ∈ B implies 0 ∈ B + v, we can apply Lemma 3.3.2 to the following (with subsequent steps similar to the ones just performed): which is equal to (3.8).

Mixing and ergodic properties
Definition 3.4.1. We say a family (F u ) u∈Z 2 of discrete random variables on the probability space (Ω, F, P) is mixing w.r.t. P if for any v ∈ Z 2 \ 0, any finite box B ⊆ Z 2 , and any realizations Theorem 3.4.2. The streetgrid SG(X) is mixing w.r.t. P .
Proof. Take B a box. Because of the stationarity of the process SG(X), we can suppose 0 ∈ B without loss of generality. Let g ∈ N 0 B . As for the shift, take v ∈ Z 2 such that v 0 > 0. The case v 1 > 0 can be proven analogously. Define the cutting-event The meaning of this event is that between B and B + nv, there is a vertical street of higher level than any of the streets in g and h.
The event C n satisfies, for n ∈ N large enough, C n also has the property to render independent events happening on B and B + nv: it implies that the smallest block around B w.r.t. SG(X) and the smallest block around B + nv w.r.t. SG(X) are disjoint, which means that different points are responsible for the two. We hence have, with the events Corollary 3.4.3. The streetgrid SG(X) is totally ergodic.

Consequences
Corollary 3.5.1. The environment ω is stationary.
Proof. This is true because in order to determine every point ω(u), u ∈ Z 2 , the same function is applied to the SG-values around u in a local and stationary manner, and because SG(X) is stationary.
Corollary 3.5.2. The environment ω is mixing.
Proof. To prove this, we would like to carry over the arguments from the proof of Theorem 3.4.2. However there is an issue about ω (as a function of SG) not being completely localized in the sense that in order to determine ω on a box B, one needs to know the width of the streets present in B. Recall that only if a street has its full planned width the biased transition probabilities are placed on it; else, the transition probabilities of a simple random walk are used. one migth want to think of this box as a thicker closure, with thickness β SG (B) . In other words, ω| B is SG(B)| B β SG (B) -measurable. We will use this fact in the following calculations. Take g, h ∈ (S 2 ) B .
Properties of the random walk 4.1 The main theorem and the idea of its proof where ω is the environment from Definition 2.3.2 with its corresponding probability measure P , and (X t , P ω 0 ) the random walk from (1.2).
A similar assertion holds with 1 replaced by − 1. The two together imply Theorem 1.0.1. Recall the heuristical description at the beginning of Subsection 3.2. The idea of the proof of the Theorem is that the random walk X t has positive probability to follow the streets in the initial grid InitGrid, at least from some starting point onwards. The starting point has positive probability to be reached directly from the origin. From there the random walk proceeds exactly like described, except for the "going straight" part: as it is a random walk, we have to take care of some fluctuations; but this is possible thanks to the streets growing nicely, see Corollary 3.2.7.

Definitions and Lemmata
The reason for the restriction to m ≥ 5 is the same as in (3.2).
Lemma 4.2.3. Take some sequence (n m ) m≥5 such that βm 2 ≤ n m ≤ β α m+2 , m ≥ 5. Also take a sequence of starting points v m ∈ S a m , m ≥ 5. We consider the (non-random) environment defined by setting a (u) := ω 0 if u 0 ≤ 0, ω 0 if u 0 > 0 for all u ∈ Z 2 . It engenders the random walk (X t ) t≥0 in the environment a , starting in v m , given by the measure P a vm . It now holds that P a vm (τ E a m (nm) < τ T a m (nm) ) is summable in m, where τ · is from Definition 4.2.1. linear speed of X ·,1 is at least the same as the one of X ·,0 . With the box growing large, even if X · starts at the easternmost possible point βm 16 e 0 , by the time X ·,1 reaches βm 16 , X ·,0 will not have reached 3βm 16 + 1. • to the horizontal piece of ∂B a in the northern west, which is exponentially becoming unlikely because the drift to the north provides that the probability of X ·,1 being smaller than 0 at the time X ·,0 hits βm 16 is decaying fastly.
Lemma 4.2.6. P ω vm (τ E c m < τ T c m ) is summable in m for any arbitrary sequence v m ∈ S c m , m ≥ 5.
Lemma 4.2.7. Take some sequence (n m ) m≥5 such that βm 2 ≤ n m ≤ β α m+2 , m ≥ 5. Also take a sequence of starting points v m ∈ S A m , m ≥ 5. Define the environment by setting The arguments needed for the proofs of these last four Lemmata are the same as in the two preceeding proofs, which is why we omit them here.

Proof of the Theorem
We prove Theorem 4.1.1 by showing that the random walk has positive probability to hit a certain sequence of target sets leading to infinity in a prescribed order, while not hitting the succession of escape-sets we define at the same time. The sets will be based on the ones who have just been treated in the Lemmas 4.2.3, 4.2.5, 4.2.6, 4.2.7, 4.2.8, and 4.2.9.  Figure 10: Target areas. The path has positive probability to hit them in that order.
Because of the ellipticity of the random environment, and because M from (3.2) is P -a.s. finite, the first probability on the right hand side is strictly larger than 0. The product being larger than 0 is thus equivalent to †∈ABC m≥M +1 The case "=" cannot occur because the target-and escape-sets are disjoint. Hence, what we need to show is the P -almost sure summability in m of Let us look at the case † = "b". Note that where b is the one defined in Lemma 4.2.5. This is true because of the placements of O b m , and Corollary 3.2.7. So, the probability is the same as the one in Lemma 4.2.5, which yields summability. The other cases in ABC can be treated the same way using Lemmas 4.2.3, 4.2.6, 4.2.7, 4.2.8, and 4.2.9; for "a" and "A", we need to remark that (n a m ) m and (n A m ) m satisfy the necessary conditions.