Quantitative estimates for the flux of TASEP with dilute site disorder

We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. For high dilution, we obtain an accurate description of the flux. The result is established undera decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum. To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.


Introduction
The flux function, also called current-density relation in traffic-flow physics [12], is the most fundamental object to describe the macroscopic behavior of driven lattice gases. The paradigmatic model in this class is the totally asymmetric simple exclusion process (TASEP), where particles on the one-dimensional integer lattice hop to the right at unit rate and obey an exclusion rule. Density ρ ∈ [0, 1] is the only conserved quantity and is associated locally with a flux (or current) that is defined as the amount of particles crossing a given site per unit time in a system with homogeneous density ρ. For TASEP, the flux function is explicitly given by f 0 (ρ) = ρ(1 − ρ). (1.1) Flux of TASEP with site disorder probability 1 − ε, in which case it has rate 1, or "slow" with probability ε, in which case its rate has some distribution Q with support (r, 1] for some r ∈ (0, 1). Under some assumption on the distribution Q, and for sufficiently diluted disorder, i.e. ε small enough, we prove (Theorem 2.4) the existence of a flat segment and determine the limiting size of this segment when ε vanishes. Moreover, we prove (Theorem 2.6) the convergence of the whole flux function to an explicit function, which exhibits a sharp transition at ε = 0. We stress the fact that Sly's argument [38] does not require any assumption on Q nor on the dilution of the disorder, however the control on the flux in [38] is less precise for small ε than the one we provide in Theorems 2.4 and 2.6. The reason for this is that our renormalization approach is a perturbation of ε = 0, while Sly's approach involves a comparison with the homogeneous TASEP corresponding to ε = 1. It follows that Sly's estimate on the size of the flat segment, is (unlike ours) not optimal for small ε.
The physical interpretation of the flat segment in the flux [24] is the emergence at different scales of atypical disorder slowing down the particles and leading to traffic jams. As one moves ahead along the disorder, slowest and slowest regions are encountered, with larger and larger stretches of sites with the minimal rate r (or near this minimal rate). Locally a slow stretch of environment inside a typical region is expected to create a picture similar to the slow bond TASEP introduced in [23]. It is known that a slow bond with an arbitrarily small blockage [8] restricts the local current. On the hydrodynamic scale [36], this creates a traffic jam with a high density of queuing vehicles to the left and a low density to the right, that is an antishock for Burgers' equation. Renormalization turns the problem into a hierarchy of slow-bond like pictures, where at each scale, the difference between the "typically fast" and "atypically slow" region becomes smaller and smaller. Slower jams will gradually absorb faster ones so that one expects to see a succession of mesoscopically growing shocks and antishocks. Some results in this direction were obtained in [20] in the case of particle disorder. Even though a single slow bond induces a phase transition, it is not clear if the transition will remain in presence of disorder or if the randomness rounds it off as in equilibrium systems [2].
Renormalization is often key to analyze multi-scale phenomena in disordered systems; we refer to [40,43] for a general overview. Our renormalization scheme controls rigorously the multi-scale slow bond picture described in the paragraph above. A major difficulty compared to the single slow bond is that as one moves to larger scales, the typical maximum current associated with a given scale and the maximum current associated with the rare slow regions occurring at the same scale converge to the same value r/4 (with r the minimal value of the jump rates). Thus a delicate issue is to show that this small current difference exceeds the typical order of fluctuations at each scale, so that the slow-bond picture remains valid at all scales. To quantify this difference, we rely on an assumption on the decay of the maximum current in a finite box (2.18). This assumption is satisfied under a condition on the tail of the rate distribution Q near its minimum r (see Lemma 2.3). Heuristics suggest that this assumption should be always valid although we have not been able so far to prove this conjecture.
We achieve our renormalization scheme by formulating the problem in wedge LPP framework with columnar disorder and exponential random variables. In the LPP framework, the phase transition takes the form of a pinning transition for the optimal path [26]: the path gets a better reward from vertical portions along slow parts of the disorder. The core of this approach is to obtain a recursion between mean passage times at two successive scales. Like many shape theorems [28], our results partially extends to LPP with more general distributions.
Another interpretation of our renormalization scheme (see [7,Section 3.3.2]) is that it consists in a hierarchy of homogenization problems for scalar conservation laws which approximate the particle system in blocks of mesoscopic size. As explained in [7], the homogenization of a one-dimensional scalar conservation law with a "fast" flux and a "slow" flux is easily seen to produce a flat segment as long as the fluxes in each block are bell-shaped (but not necessarily concave). With this bell-shape assumption on the flux function, the emergence of antishocks, previously established in [36] for TASEP with a slow bond, was shown (see [4]) to hold in more general asymmetric models with single localized blockage, even in the absence of mapping on a percolation problem. Therefore, our renormalization picture suggests that the emergence of the flat segment should be true for such exclusion-like models with bell-shaped flux. However, we are currently far from being able to implement these ideas microscopically in the absence of a LPP representation. The main reason is that the latter yields fluctuation estimates on the current, which makes the slow-bond (or homogenization) picture effective even when the difference between slow and fast cells tends to zero. Even though the implementation of the renormalization is model dependent, we stress the fact that our renormalization strategy should be useful to study other dynamics in random media. In particular, it was implemented in [10] to control the velocity of interfaces moving in a disordered environnement.
The paper is organized as follows. In Section 2, we set up the notation and state our main result. In Section 3, we formulate the problem in the last passage percolation framework and introduce the reference flux and the passage time functions. In Section 4, we introduce the renormalization procedure and describe the main steps of the proof. In Section 5, we prove a recurrence which links the passage time bounds of two successive scales. This is the heart of the renormalization argument. In Section 6, we study this recurrence in detail and show that it propagates the bounds we need from one scale to another. In Section 7, we establish an important fluctuation estimate needed in Section 5. Finally, the proofs of our main theorems are completed in Section 8.

TASEP with site disorder
Let N := {0, 1, . . .} (resp. N * := {1, 2, . . .}) be the set of nonnegative (resp. positive) integers. The disorder is modeled by α = (α(x) : x ∈ Z) ∈ A := [0, 1] Z , an i.i.d. sequence of positive bounded random variables. The precise distribution of α will be defined in Section 2.2. For a given realization of α, we consider the TASEP on Z with site disorder α. The dynamics is defined as follows. A site x is occupied by at most one particle which may jump with rate α(x) to site x + 1 if it is empty. A particle configuration on Z is of the form η = (η(x) : x ∈ Z), where for x ∈ Z, η(x) ∈ {0, 1} is the number of particles at x. The state space is X := {0, 1} Z . The generator of the process is given by for any function ϕ on X depending on finitely many sites (the set of such functions, called cylinder functions, is a core for the generator L α ), where η x,x+1 = η − δ x + δ x+1 denotes the new configuration after a particle has jumped from x to x + 1, and δ x is the configuration that is empty outside site x and has a particle at x. Current and flux function. The macroscopic flux function f can be defined as follows. For η ∈ X, we denote by J α x (t, η) the rightward current across site x up to time t, that is the number of jumps from x to x + 1 up to time t, in the TASEP (η α t ) t≥0 with generator (2.1) starting from initial state η. For ρ ∈ [0, 1], let η ρ be an initial particle configuration Flux of TASEP with site disorder with asymptotic particle density ρ in the following sense: We then set f (ρ) := lim where the limit is understood in probability with respect to the law of the quenched process. It is indeed shown in [35] that the function f in (2.3) exists for almost every realization of the disorder α, and does not depend on the latter, nor on the choice of initial configurations η ρ satisfying (2.2). Other definitions of the flux and the proof of their equivalence with the above definition can be found in [7].
It is shown in [35] that f is a concave function, see (3.9) below. It was conjectured in [42] that for i.i.d. disorder, the flux function f exhibits a flat segment, that is an interval Figure 1). The proof of [38] uses a comparison with a homogeneous rate r TASEP. We introduce a different approach, based on renormalization and homogenization ideas, viewing the disordered model as a perturbation of a homogenous rate 1 TASEP. This yields (see Theorems 2.4 and 2.6 below) not only an independent proof of the existence of a flat segment, but also optimal estimates when the density of defects is small enough.

The flux and flat segment for rare defects
From now on, we consider i.i.d. disorder such that the support of the distribution of α(x) is contained in [r, 1], where r ∈]0, 1[ is the infimum of this support. Then, as stated in the following proposition, the flux is bounded from above by r/4. This result comes from the fact that the current of the disordered system is limited by atypical large stretches with jump rates close to r. On these atypical regions, the system behaves as a homogeneous rate r TASEP which has maximum current r/4. A detailed proof can be found in Appendix A. For our main results, we formulate additional assumptions on the distribution of the environment. We assume that the disorder is a perturbation of the homogeneous case with rate 1. Let Q be a probability measure on [r, 1], such that r is the infimum of the support of Q. Given ε ∈ (0, 1) a "small" parameter, we define the distribution of α(x) by Flux of TASEP with site disorder Let us denote by f ε the flux function (2.3) for the disorder distribution Q ε . We then define the edge of the flat segment as (2.5) It follows from Proposition 2.1 that ρ c (ε) ≤ 1/2. It is also known (see [35]) that f ε is symmetric with respect to ρ = 1/2, i.e.
Therefore, (2.5) is equivalent to saying that the flat segment of f ε is the interval [ρ c (ε), 1− ρ c (ε)]. The following monotonicity properties with respect to ε can be established (see Appendix B) by standard coupling arguments.
Proposition 2.2. The macroscopic parameters are monotone with respect to the dilution: Our main results (Theorems 2.4 and 2.6 below) hold under a general assumption (H) on the disorder distribution Q which will be stated and explained in the next subsection. Concretely, as will be shown there, assumption (H) is easily implied by the following simple tail assumption: Assumption (H) holds if the following condition is satisfied: Theorem 2.4. Under assumption (H), there exists ε 0 > 0 such that ρ c (ε) < 1 2 for every ε < ε 0 . Furthermore, the size of the flat segment is explicit when ε vanishes: (2.9) Remark 2.5. It follows from (2.8) that the limiting value of the length 1 − 2ρ c (ε) of the flat segment is √ 1 − r. The result of [38] is that ρ c (ε) < 1/2 for any ε ∈ (0, 1), without requiring assumption (H) or (2.7). The proof of [38] yields the upper bound When the expectation in (2.10) is computed from the disorder distribution (2.4), it yields (we now denote it with an index ε to emphasize its dependence on ε) which converges to 1 as ε → 0 to 0. Thus, in the dilute limit, the quantity µ = µ ε in (2.10) converges to (1/r) − 1, and the upper bound on ρ c (ε) converges to (1 + r)/4, which is strictly bigger than ρ c (0) in (2.8). Correspondingly, in the dilute limit, (2.10) gives a lower bound (1 − r)/2 on the length of the flat segment, which is strictly smaller that √ 1 − r. f ε (ρ) = f 0 (ρ). (2.12) Remark that f 0 = f 0 , the latter flux function corresponding to the case ε = 0, that is a homogeneous rate 1 TASEP: recall that f 0 (ρ) := ρ(1 − ρ), as defined in (1.1). The fact that the limit in (2.12) is f 0 and not f 0 is the sharp transition announced in the introduction. It can be understood as follows: between highly dilute defects, the system is a homogeneous rate 1 TASEP. However, the memory of the defects persists (only) through the maximum flux value r/4 instead of 1/4 = max ρ f 0 (ρ).
It is important to note that, although ρ c (0) is the lower bound of the flat segment of f 0 , the convergence (2.8) is not a direct consequence from (2.12). Theorem 2.6 does not imply the existence of the flat segment for given ε either. However, the proofs of (2.8) and (2.12) are closely intertwined and both follow from our renormalization approach.

A general assumption
Let us now state assumption (H) which is used in Theorems 2.4 and 2.6. For this we first define the maximal current in a finite domain.
In the following, α B := (α(x) : x ∈ B) denotes the environment restricted to B. Consider the TASEP in B with the following boundary dynamics: a particle enters at site x 1 (if x 1 > −∞) with rate α(x 1 − 1) if this site is empty; a particle leaves from site x 2 (if x 2 < +∞) with rate α(x 2 ) if this site is occupied. Note that this process depends on the disorder in the larger box where η ± δ x denotes the creation/annihilation of a particle at x. If x 1 = −∞ or x 2 = ∞, the corresponding boundary term does not exists in (2.14). When B is finite, this process has a unique invariant measure. This allows the following definition of the maximal current for the disordered TASEP restricted to B.
where ν α B # is the unique invariant measure for the process on B with generator L α B # .

Remark 2.8.
One can see that the right-hand side of (2.15) is independent of x by writing that the expectation under ν α B # of L α B # η(x) for x ∈ [x 1 , x 2 ] ∩ Z (which yields the difference of two consecutive integrals in (2.15)) is zero.
To simplify notation, we shall at times omit the dependence on α B # and write j ∞,B # . It is well-known [14] that in the homogeneous case, i.e. when α(x) = r for all x in [0, N ] (with r a positive constant), j ∞,[0,N ] is no longer a random variable and (2. 16) In fact, explicit computations [14] show that, for some constant C > 0,

Assumption (H).
There exists b ∈ (0, 2), a > 0, c > 0 and β > 0 such that, for ε small enough, the following holds for any N : Note that if assumption (H) is satisfied for some b ∈ (0, 1), it is satisfied a fortiori for b = 1. Thus, from now on, without loss of generality, we will assume that b ∈ [1, 2). We stress the fact that the condition b < 2 is borderline as a simple comparison with the homogeneous case (2.17) leads to a control of the decay for b = 2.
We have not been able to prove that assumption (H) is satisfied for Bernoulli disorder for some well chosen parameters a > 0 , c > 0, b ∈ (0, 2), β > 0. This follows from elementary computations.

Last passage percolation approach
The derivation of Theorems 2.4 and 2.6 relies on a reformulation of the problem in terms of last passage percolation.

Wedge last passage percolation
Let Y = (Y i,j : (i, j) ∈ Z × N) be an i.i.d. family of exponential random variables with parameter 1 independent of the environment (α(i) : i ∈ Z). In the following, these variables will sometimes be called service times, in reference to the queuing interpretation of TASEP. The distribution of Y is denoted by IP and the expectation with respect to this distribution by IE. Let Index i represents a site and index j a particle. Given two points (x, y) and (x , y ) in Z × N, we denote by Γ((x, y), (x , y )) the set of paths γ = (x k , y k ) k=0,...,n such that (x 0 , y 0 ) = (x, y), (x n , y n ) = (x , y ), and (x k+1 − x k , y k+1 − y k ) ∈ {(1, 0), (−1, 1)} for every k = 0, . . . , n − 1. Note that Γ((x, y), (x , y )) = ∅ if (x − x, y − y) ∈ W. Given a path γ= (x k , y k ) k=0,...,n ∈ Γ((x, y), (x , y )), its passage time is defined by The last passage time between (x, y) and (x , y ) is defined by We shall simply write T α (x, y) for T α ((0, 0), (x, y)). This quantity has the following particle interpretation.
Then (η α t ) t≥0 is a TASEP with generator (2.1) and initial configuration η * = 1 Z∩(−∞,0] , and H α is its height process. Besides, if we label particles initially so that the particle at x ≤ 0 has label −x, then for (x, y) ∈ W, T α (x, y) is the time at which particle y reaches site x + 1. Let us recall the following result from [35].
is well-defined in the sense of a.s. convergence with respect to the distribution of Y . It is finite, positively 1-homogeneous and superadditive (thus concave). The function is well-defined in the sense of a.s. convergence with respect to the distribution of Y . It is finite, positively 1-homogeneous and subadditive (thus convex). These functions do not depend on α and are related through h(t, x) = inf{y ∈ [0, +∞) : τ (x, y) > t}, (3.5) τ (x, y) = inf{t ∈ [0, +∞) : h(t, x) ≥ y}.
for some convex function k : R → R + . It is known that for homogeneous TASEP (that is (3.8)

Reformulation of Theorems 2.4 and 2.6
In this section, we are going to rewrite the flux in the last passage framework and show that Theorem 2.4 can be deduced from a statement on the passage time. It is shown in [35] that the macroscopic flux function f is related to k (defined in (3.7)) by the convex duality relation which implies concavity of f . We now introduce a family of "reference" macroscopic flux functions and associated macroscopic passage time and height functions. Let 0<ρ c ≤ 1/2 and J ≥ 0. For ρ ∈ [0, 1], we define (see Figure 1) proving the existence of ε 0 > 0 and ρ ∈ [0, 1/2) such that the flux remains above r/4 for densities in [ρ, 1 − ρ]: As f ε ≤ r/4 by Proposition 2.1, lower bound (3.11) implies that f ε equals r/4 on [ρ, 1 − ρ]. Since f ε is concave and symmetric (2.6), ρ c (ε) in (2.5) is characterized by: The convex conjugate of f ρc,J through Legendre duality (3.9) is defined for x ∈ R by Finally, one can associate to k ρc,J a passage time function and a height function, related by (3.5)-(3.6), and defined for x ∈ R and y ≥ x − by where x + = max{x, 0} and x − = − min{x, 0}. It follows from (3.5), (3.7) and (3.9) that Hence, the quantity ρ c (ε) in (2.5) can be defined equivalently as follows: Thus the lower bound (3.11) on the flux can be rephrased in terms of an upper bound on the last passage time. Theorems 2.4-2.6 are consequences of the following theorems, which will be proved in the next sections. with τ ρ,r/4 defined in (3.14). In particular, τ ε (., y) has a cusp at x = 0 and the optimal value ρ c (ε) introduced in (3.16) converges in the dilute limit: where τ 0 is the counterpart of the flux function f 0 defined in (2.12) and where ρ c (0) was introduced in (2.9). Theorem 3.3 can be partially extended to LPP with general service-time distribution and heavier tails. In this case the particle interpretation is less standard, though the process can be viewed as a non-markovian TASEP (see e.g. [25] and [35]). Our approach (and the extension just explained) also applies to other LPP models with columnar disorder (in the wedge picture) or diagonal disorder (in the square picture), like for instance the K-exclusion process [35].

Last passage reformulation of assumption (H)
We will reformulate condition (H) in the last passage setting. To this end, we define y) and (x , y ) are such that x and x lie in B, we define Γ B ((x, y), (x , y )) as the subset of Γ((x, y), (x , y )) consisting of paths γ that lie entirely inside B in the sense that x k ∈ B for every k = 0, . . . , n. We then define The counterpart of Definition 2.7 is exists IP-a.s. for x 0 ∈ B, does not depend on the choice of x 0 , and defines a random variable depending only on the disorder restricted to B. Besides, if B is finite, we have ) in the open system restricted to , (x 0 , m)), by Remark 3.2 above, T α B ((x 0 , 0), (x 0 , m)) is dominated stochastically by the sum of 2m i.i.d. E(1) random variables. Thus, as in Theorem 3.1, the limit in (3.22) also holds in L 1 .
Note that in (2.15), j ∞,B # (α B # ) was defined as the maximum current for the TASEP in B. By (2.13) and (3.24), (B ) # = B so that the above lemma is consistent with (2.15).
The proof of Lemma 3.5 is postponed to Appendix C.
To simplify notation, we shall at times omit α B and write T ∞,B , j ∞,B . We can now restate condition (H) in terms of last passage time: Assumption (H). There exists b ∈ (0, 2), a > 0, c > 0 and β > 0 such that, for ε small enough, one has for any N ∈ N * : The constants a, c in (2.18) are different from those in (3.25), but b and β are the same.

Renormalization scheme
From now, we are going to focus on the last passage percolation model in order to prove Theorems 3.3 and 3.4. We first describe a renormalization procedure to show that a bound of the form (3.17) holds with high probability at every scale (see Proposition 4.2 below).

Definition of blocks
Let n ∈ N * be the renormalization "level" and K n = K n (ε) the size of a renormalized block of level n (by block we mean a finite subinterval of Z). For n = 1, we initialize K 1 = K 1 (ε) and define a block B of order 1 to be good if it contains no defect, i.e. α(x) = 1 for every x ∈ B. Otherwise, the block is said to be bad.
with γ ∈ (0, 1). For n ≥ 1, a block B of order n + 1 has size K n+1 and is partitioned into l n disjoint blocks of level n whose size is K n . This block is called "good" if it contains at most one bad subblock of level n, and if condition (4.2) below on the maximum current in the block holds: where the constants a, b were defined in (2.18). Otherwise B n+1 is said to be bad. We stress the fact that the status (good or bad) of B n+1 depends only on the disorder variables α Bn+1 in B n+1 and not on the exponential times Y i,j .
The renormalization is built such that large blocks are good with high probability.
Let q n =q n (ε) denote the probability under P ε that the block [0, K n − 1] ∩ Z, at level n, is bad. Since the distribution of α is invariant with respect to space shifts, q n (ε) is also, for any x ∈ Z, the probability that [x, x + K n − 1] ∩ Z is a bad block. In the rest of this paper, quantities K n , l n , q n will be written with or without explicit dependence on ε, depending on necessity.
Indeed, the first inequality is a union bound over the K 1 sites of the block for the probability ε of each site being a defect. The second inequality is a union bound over the l n (l n − 1)/2 ≤ l 2 n pairs of subblocks for the probability q 2 n of both subblocks being bad at order n. The last term ζ n+1 estimates from above the probability that the maximum current in the whole block does not satisfies (4.2) (see (2.18) and (3.25)).
On the one hand, q 1 ≤ 2ζ 1 follows from K 1 ≤ K * (ε). On the other hand, ζ n+1 ≥ 4l 2 n ζ 2 n is equivalent to Assuming 0 ≤ γ < β β+2 , since K n is increasing in n, the above inequality holds for all n ≥ 1 if it holds for n = 1, which is equivalent to

Mean passage time in a block
The strategy to prove Theorem 2.4 is now as follows. To each block B = [x 0 , x 1 := x 0 + K n − 1] ∩ Z of level n, we associate finite-size macroscopic restricted passage time functions (in the left and right directions) taking as origin either extremity of the block: which depend only on the disorder α B . To keep compact notation, we will write both functions in the form τ α B (σ, y) with σ = ±1 and y ≥ σ − = − min{σ, 0}.
The main step towards Theorem 3.3, stated in Proposition 4.2 below, is to prove that the mean passage time at each level n remains bounded by the reference function (3.14) with parameters ρ n , J n appropriately controlled to ensure that the flat segment is preserved at each order.
such that: (i) Uniformly over good blocks B at level n and for every σ ∈ {−1, 1} (ii) lim n→∞ J n = r/4 and J n > r/4 for all n ∈ N * , (iii) lim sup n→∞ ρ n < 1/2 and with the definition (2.9) of ρ c (0) Once Proposition 4.2 is established, completing the proof of Theorem 3.3 (and thus Theorem 2.4) is a relatively simple task, which boils down to obtain a similar bound on unrestricted passage times (see Section 8). The upper bound τ ρn,Jn is the counterpart, in the last passage percolation setting, of the modified flux f ρn,Jn depicted Figure 1.
The derivation of Theorem 2.6 relies on a refined version of (4.7) in the dilute limit.

Coarse-graining and recursion
The strategy of the proof of Propositions 4.2 and 4.3 is based on a coarse-graining procedure. We will first show a general estimate for any good block B at level n ≥ 1 and σ ∈ {−1, 1}: (4.10) Then, the sequence (g n ) n≥1 defined on [σ − , +∞) by and g n+1 (σ, y) := sup satisfies the bound (4.9) for any good block B and n ≥ 1.
The proof of this proposition is postponed to Section 5. The recursion (4.12) between g n and g n+1 is obtained by decomposing a path at level n + 1 into subbpaths contained in subblocks of size K n . We then express the total passage time as a maximum of a sum of the partial passage times in each subblock, where the maximum is over all possible intermediate heights of the path at the interfaces. The term δ n ϕ on the last line of (4.12) is a fluctuation estimate (see Proposition 5.1 below) on the difference between the expectation of the maximum of partial times and the maximum of the expectations. The first line of the r.h.s. of (4.12) comes from approximating each partial passage time with its mean and using the induction hypothesis (4.9).
Assumption (H) ensures that the decay of j n to r/4 is slow enough so that the additional fluctuations of order ∆ n do not hinder property (ii) of Proposition 4.2. Implicit in the statement of Proposition 4.5 is the parameter γ ∈ (0, 1) defined by (4.1), which regulates the speed of growth of our renormalization blocks. Recall that (4.3) requires γ small enough for good blocks to be typical.
For Proposition 4.5 to be useful, γ has to be chosen possibly even closer to 0, so that the upper bound (4.17) vanishes in the limit n → ∞. This is the content of the next proposition, which will imply Proposition 4.2.
The following result established in Section 6 will lead to Proposition 4.3. (4.18) Recall that g n depends on ε through the coarse graining scale.

Proof of Proposition 4.4
In this section, we prove the recursion in Proposition 4.4. To this end, we decompose a path of length K n+1 according to its traces on the interfaces between the subblocks of size K n (see Figure 2). The set of such traces will hereafter be called the "skeleton" of the path. The idea is to use (4.9) as an induction hypothesis for the subpaths in each block of size K n . If we neglect the fluctuations of these subpaths, the "'mean" computation reduces to optimizing the positions of the traces so as to maximize the total passage times of subpaths of level n. This "mean" induction relation is altered by an error term (see Proposition 5.1 below) arising from fluctuations of the subpaths as well as the entropy induced by the many possible skeletons.
LetΓ n ((x, 0), (x , y )) denote the set of skeletons of all paths γ restricted to B connecting (x, 0) and (x , y ), that is the set of sequencesγ = (ỹ i ,z i ) i=1,...,ln ∈ (N 2 ) {1,...,ln} satisfying (5.3), with the constraintỹ i ≥ K n in the case x < x. We will simply writeΓ n when the endpoints are obvious from the context.

Passage time decomposition
Let σ = ±1 denote as in (4.6) the direction of the paths. To encompass both cases σ = ±1, we will use the following simplifying convention: an interval can be written [a, b] even if a > b, in which case it actually means [b, a]. From now on, for notational simplicity, we consider the block B = [0, σ(K n+1 − 1)], instead of a block with arbitrary position x ∈ Z. For l ∈ {1, . . . , l n }, we denote by B l := [σ(l −1)K n , σ(lK n −1)]∩Z the l-th subblock of level n in the decomposition of B. For a path skeletonγ = (ỹ l ,z l ) l=1,...,ln ∈Γ n , definẽ if i ≥ 2 andh 1 = 0. The quantityh i represents the height at which a path with skeletonγ enters block i without ever returning to block i − 1. For a path γ ∈ Γ B ((0, 0), (σ(K n+1 − 1), y )) with skeletonγ, we have that where U α B (γ) is the contribution of the horizontal crossings in the blocks B l and V α B (γ) the contribution of the vertical paths at the junction of the blocks B l (see Figure 3).
Noticing that the second inequality in (5.4) is an equality if and only ifγ is the skeleton of the optimal path, we get that is the "mean optimization problem" and a "fluctuation part" defined for y = K n+1 y as F n (y) = IE 1 K n+1 max γ∈Γn((0,0),(σ(Kn+1−1),y )) The term (5.8), which involves known information from subblocks, will give the main recursion structure, while (5.9) will be an error term. The latter will be controlled by fluctuations and entropy of paths. The precise result that will be established in Section 7 is the following: Proposition 5.1. With the notation (5.9), one has uniformly in y F n (y) ≤ δ n σ 2 + y 1 + log(1 + y) 3/2 , (5.10) with δ n defined in (4.10).
We will in fact replace the upper bound in (5.10) by a slightly worse one for the sole purpose of making it a concave function of y, which is important for us. We therefore observe that F n (y) ≤ G n (y) := δ n ϕ(y), (5.11) where ϕ is the function defined by (4.13). The concavity of ϕ can be checked by a straightforward, but tedious computation.

The main recursion (4.12)
Using the skeleton decomposition, we are now going to derive Proposition 4.4. Let us explain the choice (4.11) of g 1 in Proposition 4.4. To initiate the induction relation, we need a bound at level 1 for ρ 1 and J 1 . For n = 1, a good block at level 1 contains only rates α(x) = 1. Since the restricted passage times are smaller than the unrestricted ones, and the latter are superadditive, the asymptotic shape (3.7)-(3.8) of the homogeneous last passage percolation yields the exact upper bound τ α 1,B (σ, y) ≤ √ σ + y + √ y 2 =: g 1 (σ, y).
Suppose now that the inequality (4.9) τ α n,B (σ, y) ≤ g n (σ, y) holds at step n and that g n is concave. We will show that the recursion is valid at step n + 1 with g n+1 defined as in (4.12).
We first focus on the mean optimization problem ( (recall that j n+1 was introduced in (4.2) as one of the conditions defining a good block). As B is a good block, the subblocks B l are good for all values of l = 1, . . . , l n except for possibly one bad subblock with index i 0 . The recurrence hypothesis (4.9) at level n implies that the mean passage time on a good subblock B l is bounded by IE U α B,l (σ,γ) = K n τ α n,B l σ,ỹ l K n ≤ K n g n σ,ỹ l K n . (5.14) For the possibly remaining value i 0 such that B i0 is a bad block, we use a crude upper bound by artificially extending the path in order to compare its cost to the one of a vertical connection: which yields, as in (5.12), Note that if there is no bad subblock, we will still apply (5.15) to an arbitrarily chosen subblock to avoid distinguishing this seemingly better case, which ultimately would not improve our result. Combining the above expectation bounds, we obtain n+1 (σ, y,γ), where the value ofȳ in (5.19) has been replaced by a supremum. To bound from above τ α n+1,B (σ, y) (see (5.7)), it is enough to combine (5.21) and Proposition 5.1. This completes the proof of Proposition 4.4.

Consequences of the main recursion
In this section, we prove Propositions 4.5, 4.6 and 4.7.

Proof of Proposition 4.5
As g 1 (σ, .) is concave, the recursion (4.12) implies that g n (σ, .) is a concave function for all n. For notational simplicity, we shall write details of the proof for σ = 1. In this case, we simply write g n (.) for g n (σ, .) and ρ n for ρ σ n . We will only briefly indicate what changes are involved for σ = −1. We consider the sequence (g n ) n≥1 given by the recursion (4.12) and set y n := inf y ≥ 0 : g n (y) ≤ 1 j n+1 , where g n stands for the right derivative of the concave function. Thus, if y ≥ (1 − l −1 n )y n g n+1 (y) = 1 − 1 l n g n (y n ) − y n j n+1 + y j n+1 + 1 l n j n+1 + δ n ϕ(y), (6.2) and if y ≤ (1 − l −1 n )y n g n+1 (y) = 1 − 1 l n g n l n l n − 1 y − l n l n − 1 y j n+1 + y j n+1 + 1 l n j n+1 + δ n ϕ(y). Proof. The proof of (6.4) is split in 3 steps.
Case n > 2. We are going to prove the claim by induction. Suppose that (6.4) is valid up to rank n. To show y n+1 ≥ y n , it is enough to check that Since y n > (1 − l −1 n )y n , the above derivative is computed from the expression (6.2). Thus, using the induction hypothesis (6.4), we get for n ≥ 2 where, since K n = K 1+γ n−1 , Let us respectively denote by ψ 1 (K) and ψ 2 (K) the first and second line on the r.h.s. of (6.8). Then as K → +∞, Since for b ≥ 1 and γ > 0 we have It follows that ψ(K) > 0 for K large enough. As K 1 (ε) diverges when ε tends to 0 (see (4.4)), we have that for small enough ε, y n+1 ≥ y n ≥ (1 − l −1 n )y n holds for all n ≥ 2.
As g n+1 (y n+1 ) is given by the derivative of (6.2) and ϕ is strictly concave, we have to solve As above, (6.6) implies that, for ε small enough, a solution of (6.9) exists for all n ≥ 2. This proves the second part of the claim (6.4). The proof is similar for σ = −1.
Using Lemma 6.1, we can now complete the proof of Proposition 4.5. We must show that inequality (4.16) holds for the sequence (ρ n ) n∈N * with ∆ n satisfying (4.17). By definition (6.1) of y n , the supremum in (4.15) is reached at y n so that ρ n = j n+1 g n (y n ) − y n .

Flux of TASEP with site disorder
We are going to obtain a recursion for ρ n . To this end, consider ρ n+1 = j n+2 g n+1 (y n+1 ) − y n+1 .
For σ = −1, still writing ρ n for ρ σ n , we have ρ n − 1 = sup y≥1 j n+1 g n (−1, y) − y and we get a recursion similar to (6.10) which can be rewritten For b ≥ 1, the remainder jn+1 jn+2 − 1 can be bounded by ∆ n so that the same type of inequality is also valid for σ = −1.

Fluctuation bounds : Proof of Proposition 5.1
Proposition 5.1 is proved in this section. Preliminary estimates are stated in Subsection 7.1 and then applied in Subsection 7.2, which is the body of the proof.

Concentration estimates
We shall need a classical gaussian concentration inequality for last passage times.
In the following lemma, it is assumed that the service times Y i,j involved in the definition  N) is a vector of non negative independent random variables bounded from above by rM . Let (x 1 , y 1 ) and (x 2 , y 2 ) in Z × N be such that (x 2 − x 1 , y 2 − y 1 ) ∈ W. Then T α (x 1 , y 1 ), (x 2 , y 2 ) = IE M T α (x 1 , y 1 ), (x 2 , y 2 ) + 8M L((x 1 , y 1 ), (x 2 , y 2 ))Z, where L((x 1 , y 1 ), (x 2 , y 2 )) := (x 2 − x 1 ) + 2(y 2 − y 1 ) is the length of any path connecting (x 1 , y 1 ) to (x 2 , y 2 ), and Z is a random variable with subgaussian tail We stress the fact that Gaussian bounds on last passage times are by no means optimal in the case of exponential service times, for which more refined (but also more specific) gaussian-exponential estimates are available (see e.g. [41]). However, for our purpose, they have the advantage of being both simple and sufficient, while also extending to service distributions with heavier tails, as a result of the cutoff procedure introduced in Subsection 7.2.
The above concentration inequality will be combined with the following result, established in Appendix D.
Lemma 7.2. Let A and I be finite sets. Assume that for each a ∈ A, we have a family (Y a,i ) i∈I of independent random variables such that, for every i ∈ I, Y a,i = IE Y a,i + V a,i Z a,i , where V a,i > 0, and Z a,i is a random variable such that IP(Z a,i ≥ t) ≤ e −t 2 , for every t ≥ 0. Then where |.| denotes the cardinality, and A is a universal constant.

Path renormalization: fluctuation and entropy
We now proceed in three steps. In step one, we define a cutoff procedure for the service times Y i,j , by conditioning on their maximum, in order to replace them with bounded variables, to which the results of Subsection 7.1 apply. In step two, we apply Lemma 7.2 to passage times in subblocks. This yields for the cutoff service times a result similar to the statement of proposition 5.1, but without the whole logarithmic correction.
Finally, in step three, we remove the cutoff and use a bound on the expectation of the maximum of exponential variables, to obtain a quasi-gaussian estimate with a logarithmic correction.
To estimate the cardinality |A| of the skeletons, we need the following Lemma 7.4. For every y ∈ N, one has Proof. The number of such skeletons satisfies the inequality σ ∈ {−1, 1}, Γ n (0, 0), (σ(K n+1 − 1), y ) ≤ 2l n + y − 1 2l n − 1 . (7.10) The previous upper bound follows by noticing that choosing a skeleton amounts to choosing 2l n − 1 heights corresponding to the different renewal times to reach the total height y . In fact, when σ = 1, some of these heights can be equal ifỹ i = 0 orz i = 0 for some i ≤ 2l n − 1. Thus, the number of ways for choosing the heights is bounded by the number of ways for choosing 2l n − 1 items from a set of 2l n + y − 1 items. Estimate (7.10) is actually an equality if σ = 1. Recall the inequality (7.12) Bound (7.11) follows from Cramer's exact large deviation uppper bound. For completeness, we give a derivation of (7.11) at the end of this proof.
The Legendre transform of L is h + log 2, where h is given by (7.12). Optimizing the upper bound in (7.14) over θ yields where we used that 2 Kn+1 Kn ≥ π.
Recall that IE M on the left-hand site of (7.17) stands for the expectation with respect to IP conditioned on the maximum M B (y) = M . We can now remove this conditioning by integrating both sides of (7.17) with respect to the law of M B (y). We first write IE M B (y) = m ([yK n+1 ]K n+1 ) , (7.18) where the function t ∈ [0, +∞) → m(t) is defined as the expectation of the maximum of 1 + [t] i.i.d. exponential variables of rate 1. In particular, we have m(t) ≤ C[1 + log(1 + t)], (7.19) for some constant C > 0. Thus, after conditioning on M B (y), we obtain A simple computation shows that m [yK n+1 ]K n+1 ∆ n (y) ≤ δ n σ/2 + y 1 + log(1 + y) 3/2 , with δ n given by (4.10). Using the notation of (5.9), we get F n (y) ≤ δ n σ 2 + y [1 + log(1 + y)] 3/2 .
This completes the proof of Proposition 5.1.

Completion of proofs of Theorems 2.4 and 2.6
In this section, we complete the remaining parts in the proof of Theorem 2.4. In Subsection 8.1, we deduce from Proposition 4.2 a similar statement for unrestricted passage times (that is, when the paths are not restricted to the box defined by the endpoints). Finally, Theorem 2.4 is completed, in Subsection 8.2, using the fact that most boxes are good. The dilute limit (Theorem 2.6) is studied in Section 8.3.

Bounds on unrestricted passage times
To obtain Theorem 2.4 from Proposition 4.2, we first deduce from Proposition 4.2 the following result for unrestricted passage times, i.e. passage times obtained by maximizing over paths not bound to stay in the interval between the two endpoints (see Figure 4).
We then proceed as in Section 5 by studying the mean optimization problem (that is the maximum of the expectations of the three terms in (8.5)) and estimating the error due to this approximation.
We conclude by controlling the error thanks to Proposition 8.2, in the same spirit as Proposition 5.1.
The intervals [x N , y N ] with low rates act as bottlenecks for the particle flux. Set where (θ x ) x∈Z is the group of spatial shifts, whose action on particle configurations is defined by (θ x η)(y) = η(x + y) for every η ∈ X and y ∈ Z. Similarly, (θ x α)(y) = α(x + y) for every α ∈ A and y ∈ Z. If ϕ is a cylinder function of X, we set θ x ϕ := ϕ • θ x . If µ is a probability measure on X, θ x µ := µ • (θ x ). Finally, the action of θ x on the generator L α given by (2.1) is defined by (θ x L α )ϕ = θ x (L α ϕ) for every cylinder function ϕ on X.
The sequence (µ N ) N ∈N * of probability measures on the compact space X is tight. Let µ be one of its limit points. It follows from (A.4) that µ is shift invariant, i.e. θ x µ = µ for all x ∈ Z. We claim and prove below that µ is an invariant measure for the homogeneous TASEP, that is the process with generator (2.1) with α(x) ≡ 1. By Liggett's characterization result [18] for shift-invariant stationary measures, µ is then of the form where γ is a probability measure on [0, 1], and ν ρ is the product Bernoulli measure on X with parameter ρ. Thus We now prove that µ is an invariant measure for the homogeneous TASEP. Let g : X → R be a local function that depends on η only through sites x ∈ Z such that |x| ≤ ∆, where ∆ ∈ N. Take N large enough so that ∆ < (y N − x N )/3. Notice that the generator L α defined in (2.1) satisfies the commutation relation It follows that Flux of TASEP with site disorder The last equality follows from invariance of ν α ρ . On the other hand, for x ∈ [a N , b N ] and |y| ≤ (y N − x N )/3, τ x α(y) ∈ [r, r + ε N ]. Let L denote the generator of the homogeneous TASEP on Z, that is the one obtained from (2.1) when α(x) ≡ 1. Since holds for every local function g.

B Proof of Proposition 2.2
The proof uses the following lemma.
Lemma B.1. Let B be a nonempty interval of Z, and B # as in (2.13) Assume α and α are two environments such that α(x) ≤ α (x) for every x ∈ B # . Then where J α,B # x (t, η) denotes the rightward current across site x up to time t for the process with generator (2.14).
Proof of Proposition 2.2. Statement (ii) follows from (i) and (2.5). We now prove (i). Let U = (U x ) x∈Z be a family of i.i.d. U(0, 1) random variables, and V = (V x ) x∈Z be a family of i.i.d. random variables with distribution Q, independent of U . We set Thus α ε has distribution Q ε defined in (2.4), and since V x ≤ 1, where the inequality is in the sense of product order. Let ρ ∈ [0, 1] and η ρ ∈ X satisfying (2.2). By (2.3), for almost every realization of (U, V ), the following limit holds in probability f ε (ρ) = lim The conclusion then follows from (B.3) and Lemma B.1.
Proof of Lemma B.1. Statement (ii) follows from (i), Definition 2.7 and the fact that j ∞ = lim t 1 t J α,B # (t, η). We now prove (i). Let B = [x 1 , x 2 ] ∩ Z, with x 1 , x 2 ∈ Z such that x 1 ≤ x 2 . We are going to couple the TASEP's (η α t ) t≥0 and (η α t ) t≥0 in environments α, α starting from the same configuration η ∈ {0, 1} B . If x 1 = −∞, the initial configuration η is extended so that site x 1 − 1 has an infinite stack of particle, and there is no particle to the left of x 1 − 1. Particles of the extended initial configuration are labelled increasingly towards the left. We choose the initial labeling so that the lowest label of a particle in the stack is 1. Thus the set of labels is The coupled evolution is defined using a Harris type construction, as in [21], of both TASEP's from a common Poisson point measure on (0, +∞) × Z × (0, 1) with intensity dtdx1 (0,1) (u)du, where dx is the counting measure on Z. If (t, x, u) is a Poisson point, and β ∈ {α, α }, a particle jumps from x to x + 1 in the configuration η β . if one of the following holds: either x = x 1 − 1 and η β t− (x 1 ) = 0, or x = x 2 and η β t− (x 2 ) = 1, or x 1 < x < x 2 , η β t− (x) = 1 and η β t− (x + 1) = 0.
If a jump occurs from the stack, the particle that jumps is the stack particle with the lowest label. Let σ i (t), resp. σ t (i), denote the position at time t of particle i in η α t , resp. η α t . The initial conditions are identical, i.e. σ 0 (i) = σ 0 (i) for every i ∈ I. As a consequence of the assumption α ≤ α and of the rules (i)-(iii), the order between the particle configurations is preserved by the coupling at any time ∀i ∈ N, σ t (i) ≤ σ t (i).

C Proof of Lemma 3.5
Before deriving Lemma 3.5, we first explain a mapping between the restricted passage times in a box B and the TASEP restricted to B with reservoirs.

C.1 Last passage times in a finite domain
Let B := [x 1 , x 2 ] ∩ Z. The purpose of this subsection is to give an interpretation of the passage times (3.21) restricted to B in terms of an open disordered TASEP on B := [x 1 + 1, x 2 ] ∩ Z with generator L α B , see (2.14) (recall from (2.13) and (3.24) that (B ) # = B). It is convenient to view the dynamics generated by (2.14) as follows. We add an infinite stack of particles (reservoir) at site x 1 , and a site x 2 + 1 where the number of particles is not restricted. Particles enter B from the stack at x 1 , and when they leave, they stay at x 2 + 1 forever. We are going to check that T α B (x 1 , 0), (i, j) has the same distribution as the time when particle j reaches site i + 1 in the process generated by L α B , if the initial state is given by σ 0 (j) = x 1 1 {j≥0} + (x 2 + 1)1 {j≤−1} .
(C.1) where σ 0 (j) denotes the initial position of the particle with label j, and particles are numbered increasingly from right to left. In fact, we may define passage times associated with more general labeled initial configurations in B . By this we mean that σ 0 , instead of being defined by (C.1), can be any nonincreasing function σ 0 from Z to [x 1 , x 2 + 1] ∩ Z. For (i, j) ∈ B × Z, let T α B,σ0 (i, j) denote the time at which particle j reaches site i + 1.
where the r.h.s. was defined in (3.21). For notational simplicity, in the sequel of this subsection, we omit dependence on α, B and σ 0 , and write T (i, j) instead of T α B,σ0 (i, j). The position of particle j at time t, denoted by σ t (j) ∈ [x 1 , x 2 + 1], is given by 8) T (i, j) = sup{t ≥ 0 : σ t (j) ≤ i}. (C.9) The particle process (σ t ) t≥0 is equivalent to the following growing cluster process: C t := {(i, j) ∈ [x 1 , x 2 ] × Z : T (i, j) ≤ t} = {(i, j) ∈ B × Z : i < σ t (j)} with initial state C 0 =B. One can proceed as in [37] to show that both processes are Markovian and that the undistinguishable particle process (η t ) t≥0 defined by

C.2 Proof of Lemma 3.5
Step 0: proof of (3.22). Let Ω denote the set of sequences (Y i,j ) i,j∈Z×N equipped with the product σ-algebra and the product E(1) probability measure. This measure is invariant for the family of shift operators (Θ n , n ∈ N) defined on Ω by (Θ n Y ) i,j := Y i,j+n . (i) Let Y be a random variable such that IP(Y ≥ t) ≤ Ce −t 2 /V for all t ≥ 0, where C ≥ 1 and V > 0. Then, we have (ii) There exists a positive constant A such that the following holds. Let (X k ) k=1,...,n be independent random variables such that IP(X k ≥ t) ≤ e −t 2 for all t ≥ 0, and (V k ) k=1,...,n be nonnegative numbers. Then where Z is a r.v. such that IP(Z ≥ t) ≤ e −t 2 for all t ≥ 0. Proof of Lemma D.1. Assertion (i) follows from an immediate computation. To obtain (ii) we note that, for θ ≥ 0, Setting Y k = X k − √ π, we have, for θ ≥ 0, Λ(θ) := log IE e θY k ≤ log 1 + √ πθe θ 2 /4 − √ πθ.
Thus there exists A > 0 such that Λ(θ) ≤ Aθ 2 /4 for θ ≥ 0. Hence, by independence of the random variables X k , we get The estimate on the tail of Z follows by an exponential Markov inequality.
Proof of Lemma 7.2. By (ii) of Lemma D.1, for every a ∈ A, we have where Z a is a random variable satisfying IP(Z a ≥ t) ≤ e −t 2 for all t ≥ 0. On the other hand, by Cauchy-Schwarz inequality, i∈I V a,i ≤ |I| i∈I V a,i (note that (D.3) may be false with Z a instead of Z + a if Z a < 0, because the last term on the r.h.s. of (D.1) is then greater than (AV ) 1/2 Z a ). Next, for any t ≥ 0, we have It follows from (i) of Lemma D.1 that