CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZFIXED POINT

JEREMY QUASTEL AND SOURAV SARKAR

ABSTRACT. We show that under the 1:2:3 scaling, critically probing large space and time, the heightfunction of finite range asymmetric exclusion processes and the KPZ equation converge to the KPZ fixedpoint, constructed earlier as a limit of the totally asymmetric simple exclusion process through exactformulas. Consequently, based on recent results of [31],[10], the KPZ line ensemble converges to theAiry line ensemble.

For the KPZ equation we are able to start from a continuous function plus a finite collection of narrowwedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with|h(x)| ≤ C(1 +

√|x|) for some C > 0, or one narrow wedge. For non-nearest neighbour exclusions,

we are restricted at the present time to a class of (random) initial data, dense in continuous functions inthe topology of uniform convergence on compacts.

The method is by comparison of the transition probabilities of finite range exclusion processes and thetotally asymmetric simple exclusion processes using energy estimates.

Just before posting the first version of this article, we learned that, independently and at the same timeand place, Bálint Virág found a completely different proof of the convergence of the KPZ equation to theKPZ fixed point. The methods invite extensions in different directions and it will be very interesting tosee how this plays out.

1. INTRODUCTION

The Kardar-Parisi-Zhang (KPZ) equation,

∂th = λ(∂xh)2 + ν∂2xh+√Dξ, (1.1)

where ξ is space-time white noise, the distribution valued delta correlated Gaussian field 〈ξ(t, x), ξ(s, y)〉= δ(t − s)δ(x − y), was introduced by Kardar, Parisi and Zhang in 1986 [18] as an equation for arandomly evolving height function h ∈ R which depends on position x ∈ R and time t ∈ R+. λ, ν andD are physical constants. Its derivative u = ∂xh satisfies the stochastic Burgers equation

∂tu = λ∂xu2 + ν∂2xu+

√D∂xξ. (1.2)

A dynamic renormalization group analysis was performed on (1.2) in [12] (see also [18], [28]), predict-ing a dynamical scaling exponent z = 3/2 and strong coupling fixed point. In our language, this meansthat one expects to see a non-trivial universal fluctuation field under the 1:2:3 rescaling

h 7→ δh(δ−3t, δ−2x). (1.3)

Universal is meant in the sense that it is supposed to be the attractor as δ ↘ 0 for a huge class of relatedmodels in one dimension; discrete growth models, directed polymer free energies, driven diffusivesystems. However, it is also known that the fluctuations remember the initial state, and is therefore ascaling invariant Markov process. This process, the KPZ fixed point, was characterized in [19] throughexact formulas for its transition probabilities, based on the exact solvability of a special member of theKPZ universality class, the totally asymmetric simple exclusion process (TASEP).

Exact one point distributions have been found for the KPZ equation, and for a two way version ofTASEP known as ASEP, for a few special initial data (narrow wedge [25],[1],[23], [11], half-Brownian[26], [8],[17], Brownian [27], [16],[4], and a conjectural formula for flat [5], see also [20]). However, itis not known at the present time, nor really expected, that the solvability extends to general initial data,or to multipoint distributions (excepting perhaps the narrow wedge case.) On the other hand, it waswell understood at a physical level that the 1:2:3 scaling limit of AEP and the KPZ equation should

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CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 2

coincide with that of TASEP. Note that such a statement contains several well known conjectures suchas the convergence of multipoint distributions to those of the Airy process for narrow wedge initialdata, and (following recent results [31],[10]) the convergence of the KPZ line ensemble to the Airy lineensemble. The gap was the lack of any analytic methods to prove such a convergence; all one knewhow to do was find an exact formula and take its limit.

The main contribution of this article is an analytic method which compares the transition probabilitiesof any finite range asymmetric exclusion process to TASEP, with a close enough comparison that itshows they have the same 1:2:3 scaling limit. In the weakly asymmetric limit it also allows one to provethat the KPZ equation transition probabilities, for very general initial data, converge to the KPZ fixedpoint transition probabilities.

The key step is an estimate of the difference of the transition probabilities of the two processes, aslong as one starts with a bound on the L2 norm of the Radon-Nikodym derivative with respect to aglobal equilibrium. The second step of the proof extends the result from such initial data, to a broaderclass. In the case of the KPZ equation, key tools available are a positive temperature version of thevariational formulation and Brownian Gibbs property, and skew-time reversibility,

P(h(t, x) ≤ g(x), ∀x | h(0, ·) = f(·)

)= P

(h(t, x) ≤ −f(x),∀x | h(0, ·) = −g(·)

). (1.4)

In the case of exclusion processes, the first two are missing, while skew-time reversibility holds forASEP but not in general for AEP. This is because the order of height functions is only preserved in thenearest neighbour case. Therefore the class of initial data for which we can prove the convergence forAEP at this time, while dense in a certain sense, is not as complete as one would wish. For the detailedresults, see Theorem 2.1.

While we cannot prove the result for all initial data for exclusion processes at this time, to ourknowledge the results for non-nearest neighbour exclusion represent the first results of KPZ universalityfor any data for any model which is not to some degree algebraically solvable.

Two natural questions arise; 1) Can one solve the initialization problem for AEP, i.e. show thatstarting from initial data having a well defined asymptotic profile, the distribution at any positive timeis close enough to a finite energy measure, while not having changed its asymptotic profile? 2) Howgeneral is the method? Again, we do not have a satisfactory answer. Here we are perturbing off TASEP,and the argument leading up to (5.3) uses many special properties of that process, in particular theskew-time reversibility (1.4) which gives the seemingly crucial Lem. 5.1.

At the same time, another proof of KPZ universality for the KPZ equation became available [30].Both proofs use as their launchpad the observation that, for the KPZ equation, it is only necessary toprove the convergence for a dense class of initial data. Our dense class involves a bound on the L2

norm of the Radon-Nikodym derivative with respect to equilibrium, and this key part of the argument isin Sec.3–6. The results of Sec. 7 prove tightness and convergence in the uniform-on-compact topologyfor the height functions of any model for which the convergence of probabilities for certain class of setsis known (see Prop. 7.1) Once that is done, one uses known properties of the process to extend to moregeneral initial data. This is done in Sec. 8–9. These arguments are somewhat universal in the sensethat the results of Sec. 8 will hold for any model satisfying convergence for finite energy initial dataand skew time reversibility. Appx. B contains the proof of the key Lem. 5.1, representing the gradientof the TASEP transition probabilities as the joint distribution of a max and argmax. The averagingargument Lem. 5.2 was written incorrectly in an earlier version of this article, which considered onlythe KPZ equation. So it was withdrawn and is replaced by the present article which contains the sameresults but also the extension to exclusion processes. Appx. C contains the proof of the main tool toextend the results from nearest neighbour to non-nearest neighbour exclusion processes, which is thestrong sector bound of Lin Xu [32] and S.R.S.Varadhan [29].

2. MODELS AND MAIN RESULTS

We first describe the finite range asymmetric exclusion model (AEP). We consider such models on Zwith non-zero drift. There are particles on the lattice, with at most one particle per site. The particlesattempt jumps x 7→ x+ v at rate p(v) where {v ∈ Z : p(v) > 0} is finite, and, to avoid degeneracies,

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 3

is assumed to additively generate all of Z, or, in other words, the underlying single particle randomwalk is irreducible. In order for our results to hold it is necessary that∑

v

vp(v) 6= 0. (2.1)

Otherwise, the process will not move on the scales we are observing1. We can always assume it ispositive, and by multiplying all the rates by a constant (i.e. changing the time scale by a constant) wemay as well assume that ∑

v

vp(v) = 1. (2.2)

The height function is a simple random walk path h(x), x ∈ Z with h(x + 1) = h(x) ± 1. Ifh(x+ 1) = h(x) + 1 we say there is a particle at x and write η(x) = 1 and if h(x+ 1) = h(x)− 1 wesay there is no particle at x and write η(x) = 0. We can alternatively let η(x) = 2η(x)− 1 and writeh(x+ 1) = h(x) + η(x).

As we said, a particle at x attempts jumps to site x+ v at rate p(v). But the jump takes place only ifthe target site is unoccupied. This is called exclusion. All the particles are doing this independently ofthe others, and since time is continuous, there are no ties to break. Note that when the jump occurs,the height function either drops by 2 at sites x + 1, . . . , x + v, if v > 0, or increases by 2 at sitesx + v + 1, . . . , x if v < 0. The special case where p(·) is nearest neighbour is referred to as ASEP(asymmetric simple exclusion process): Simple here means nearest neighbour.

We rescale the height function by the 1:2:3 KPZ scaling

hε(t, x) = ε1/2h(ε−3/2t, ε−1x). (2.3)

The height function in (2.3) lives on εZ ⊂ R. We now have

h(x+ ε) = h(x) + u(x) with u(x) ∈ {−ε1/2, ε1/2} . (2.4)

The discrete state space Sε = collection of such functions, with h(0) ∈ ε1/2Z. These are naturallyembedded in the continuous state space S = continuous functions on R by connecting h(x) and h(x+ε)with straight lines. When we say for example, a continuous function h is initial data for AEP, we meanthat at each level ε > 0, there is an initial data hε and hε → h in the uniform-on-compact topology. Anarrow wedge on the other hand, will have the maximum slope ε−1/2.

Call µε(dh) the law of the two sided symmetric simple random walk on εZ, with steps of sizeε1/2 with h(0) = 0. Let Lebε(dh(0)) be the discrete Lebesgue measure giving mass ε1/2 to eachh(0) ∈ ε1/2Z. An h ∈ Sε is built out of h(0) and u(x), x ∈ εZ by

h(x) = h(0) + h(x), (2.5)

and the product measureνε(dh) = Lebε(dh(0))× µε(dh) (2.6)

is invariant for the process.Next we turn to the KPZ equation. The solution [3], [14], [13] is a Markov process with state space

S = continuous functions on R. More precisely, we consider the Hopf-Cole solution, the logarithm ofthe solution of the stochastic heat equation (SHE),

h = γ logZ, ∂tZ = ν∂2xZ + γ−1√DξZ . (2.7)

where γ = ν/λ. SHE is one of the few nonlinear2 stochastic partial differential equations for which asolvability theory is relatively straightforward; in fact, the solution can be written as an explicit chaosseries [21]. Note that the SHE (2.7) can be solved starting from more general initial data. In particular,a narrow wedge initial data for KPZ means to start the SHE (2.7) with a Dirac delta measure. The map

1If we call m :=∑v vp(v), it is fairly straightforward to check that the proof shows convergence of hε(2t, 2x) to

sgn(m) times the KPZ fixed point running at speed |m|. In particular, ifm = 0 the argument shows that there is no movementon the KPZ time scale.

2it is linear in the initial data, but non-linear in the noise.

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 4

h 7→ δbh(δ−zt, δ−1x) transforms the coefficients in (1.1) by (λ, ν,D) 7→ (δ2−z−bλ, δ2−zν, δ2b−z+1D),so it is enough to always consider a standardized3 KPZ equation

∂th = 14(∂xh)2 + ∂2xh+ ξ

which is taken by the 1:2:3 scaling (1.3) into what we will call KPZ,δ,

∂th = 14(∂xh)2 + δ∂2xh+ δ1/2ξ. (2.8)

In the theorem below and in the rest of the paper, convergence of measures always means convergencewhen tested against bounded continuous functions.

Theorem 2.1. (1) Suppose that f0,εdνε → dρ with ε1/4‖f0,ε‖L2(dνε) → 0 and ρ-almost surely anysample path h(·) satisfies h(x) is bounded on any compact interval and |h(x)| ≤ C(1 + |x|) forsome C > 0, then started from f0,εdνε, the AEP height function hε(t, x) := hε(2t, 2x) + ε−1t/2converges as ε ↘ 0 in distribution to the KPZ fixed point at fixed time t > 0 started from ρ in theuniform-on-compact topology. In particular, with randomized wedges as the initial conditions (seeExample 6.5), the convergence in distribution holds in the uniform-on-compact topology.

(2) Let h be any continuous function such that for some C > 0, |h(x)| ≤ C(1 + |x|1/2) for allx. Then under the same scaling the ASEP height function hε starting from a deterministic h0,ε withh0,ε → h uniformly on compacts, converges in distribution to the KPZ fixed point at fixed time t > 0started from h in that topology.

For a single narrow wedge, Theorem 1.1 of [6] gives the tightness of the ASEP height function, andtherefore we have convergence in distribution in the uniform-on-compact topology for each time t > 0to a (rescaled) Airy2 process.

For multiple narrow wedge initial data or a continuous initial data h with h(x) ≤ C(1 + |x|1/2) forsome C > 0 and all x, we have convergence of the transition probabilities into sets of the form (5.1)for any g continuous satisfying |g(x)| ≤ C(1 + |x|1/2) for some C > 0.

(3) Start KPZ,δ (2.8) with a continuous function bounded above by A(1 + |x|) plus a finite collectionof narrow wedges. Then, as δ ↘ 0, the solution, after addition of the Itô factor t/12, converges indistribution, in the topology of uniform convergence on compact subsets, to the KPZ fixed point. TheKPZ line ensemble converges to the Airy line ensemble in the same sense (see Sec. 10) .

3. DIFFERENCE OF TWO MARKOV PROCESSES

Consider two time homogeneous Markov processes ht, and gt, t ≥ 0 on a state space S which sharea common invariant measure ν. We would like to estimate the difference between their transitionprobabilities pt(h,B) := Prob(ht ∈ B | h0 = h) and qt(h,B) := Prob(gt ∈ B | g0 = h).We could also start each in a probability measure f0(h)ν(dh) and ask for the difference betweenpt(f0ν,B) :=

∫pt(h,B)f0(h)ν(dh) and qt(f0ν,B) :=

∫qt(h,B)f0(h)ν(dh). Suppose the two

processes have infinitesimal generators T and L. The transition probabilities satisfy Kolmogorov’sbackward equations ∂tpt(h,B) = Tpt(h,B), and ∂tqt(h,B) = Lqt(h,B). Here T and L are actingon the h variable. On the other hand, if we start with f0(h) the processes at time t will have probabilitydistributions ft(h)ν(dh) and ft(h)ν(dh) solving the forward equations ∂tft = T ∗ft and ∂tft = L∗ft.Thus ∂sft−s(h)ps(h,B) = −L∗ft−s(h)ps(h,B) + ft−s(h)Tps(h,B). Integrating from 0 to t andwith respect to ν, and using

∫ft(h)p0(h,B)dν = qt(f0ν,B) we obtain

pt(f0ν,B)− qt(f0ν,B) =

∫ t

0

∫ft−s(h)(T − L)ps(h,B)dν(h)ds. (3.1)

Here∫A fs(h)dν(h) =

∫Prob(gs ∈ A | g0 = h)f0(h)dν(h) = qs(f0ν,A) for measurable A, and

T − L acts on ps through the variable h.One could do various things at this point, perhaps take a supremum over measurable subsets B ∈ S

in order to compute the total variation distance. In our case, the total variation is not expected to be

3The reason for the standard is that −x2/t solves the Hopf equation ∂th = 14(∂xh)

2 so all the parabolic shifts comewithout unnecessary constants.

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 5

small, and we will be satisfied if we can prove the two probabilities are close for a wide class of fixedB. To study (3.1) it is clear we will need information about both ps(h,B), and fs, 0 ≤ s ≤ t. To learnsomething about the latter, we might assume ‖f0‖22 =

∫f20dν <∞ and study what happens to

∫f2s dν:∫

f2t dν −∫f20dν = 2

∫ t

0

∫fsL

∗fsdνds = 2

∫ t

0

∫fsL

symfsdνds := −2

∫ t

0DL(fs)ds, (3.2)

where Lsym = 12(L + L∗) is the symmetric part of L. We could do the same thing with the other

process to create another Dirichlet form, D . In the applications we will be interested in, the Dirichletforms corresponding to T and L are comparable, i.e. there is a 0 < c <∞ such that

c−1D(f) ≤ DL(f) ≤ cD(f). (3.3)

In (3.2) the non-positivity of the right hand side tells us that the integral of the Dirichlet form can becontrolled by the initial L2 norm. Together with (3.3), we have∫ t

0D(fs)ds ≤ C‖f0‖22. (3.4)

Now suppose that T − L satisfies the strong sector condition: There exists C <∞ such that∣∣∣∣∫ f(T − L)gdν

∣∣∣∣ ≤ CD(f)1/2D(g)1/2, (3.5)

then by (3.1), (3.5) and (3.4) and the Cauchy-Schwarz inequality,

|pt(f0ν,B)− qt(f0ν,B)| ≤ C∫ t

0D(ft−s)

1/2D(ps)1/2ds ≤ C‖f0‖2

√∫ t

0D(ps)ds. (3.6)

The method of attack is to use this bound, with a little bit of extra averaging (see the beginning ofSec. 5), to estimate the difference between the rescaled TASEP and, respectively, AEP or WASEP.

4. GENERATORS AND DIRICHLET FORMS

The rescaled AEP (2.3), has infinitesimal generator

Lεf = ε−3/2∑x∈εZ

∑v∈Z

p(v)∇x,vf, ∇x,vf(h) = f(hx,v)− f(h). (4.1)

Here hx,v = h unless there is a particle at x and no particle at x+ εv, in which case hx,v is the newheight function after the particle has performed the jump.

We will be comparing it (using assumption (2.2)) to the rescaled TASEP, which has generator

Tεf = ε−3/2∑x∈εZ∇x,1f. (4.2)

The operator∇x,v has a symmetrized version∇symx,v f(h) = f(hx,v,sym)− f(h), where hx,v,sym = h

unless there is a particle at x and no particle at x+ εv or no particle at x and a particle at x+ εv, inwhich cases hx,v,sym is the new height function after the particle numbers at x and x+ εv have beenexchanged.

The generators (4.1) and (4.2) are invariant with respect to the (unnormalized) measure νε defined in(2.6), and have Dirichlet forms Dp

ε (f) := −∫fLεfdνε = 1

2ε−3/2∑

x∈εZ∑

v∈Z p(v)∫

(∇symx,v f)2dνε

and

Dε(f) := −∫fTεfdνε = 1

2ε−3/2

∑x∈εZ

∫(∇sym

x f)2dνε (4.3)

respectively, where∇symx is short for∇sym

x,1 . Since hx,v = ((hx1,1)x2,1) · · ·x|v|,1, it is rather easy to seethat there is a C < ∞, depending only on p(·), such that Dp

ε (f) ≤ CDε(f). In the other direction,it is easy to construct f and p(·) so that Dp

ε (f) = 0 but Dε(f) > 0. On the other hand, under ourassumption p(·) is irreducible, it is also true there is a finite string so that hx,1 = ((hx1,v1)x2,v2) · · ·x`,v`with p(vi) > 0 for each i = 1, . . . , `. Hence

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 6

Lemma 4.1. The Dirichlet forms of TASEP and AEP are comparable (3.3) with a constant c < ∞independent of ε > 0.

The symmetric simple exclusion process (SSEP) which allows flips both ways, is defined through theDirichlet form (4.3) or its generator

Sε,δf := δε−2∑x∈εZ∇symx f (4.4)

The Dirichlet form gives δ = ε1/2, i.e.Sε := Sε,ε1/2 (4.5)

but in order to approximate the KPZ equation we will want to fix δ independent of ε. The weaklyasymmetric simple exclusion process (WASEP), allows flips both ways, with a slight asymmetry. Itsgenerator is

Lε,δ = Sε,δ + Tε. (4.6)

Going back to AEP, we can write

Lε − Tε = Mε − Sε where Mε = Lε + T ∗ε , Sε = Tε + T ∗ε . (4.7)

T ∗ε = ε−3/2∑

x∈εZ∇x,−1f is just the TASEP with time reversed (or space flipped). From (2.2), Mε isthe generator of a mean zero exclusion, i.e. it is of the form (4.1) with

∑v vp(v) = 0. The following

estimate, which is key to the non-nearest neighbour case, appeared without the heights in the thesis ofLin Xu [32] and for the environment as seen from a tagged particle by S.R.S.Varadhan (Lem. 5.2 of[29]). Our variant with the heights is basically the same as [32]. For completeness we give a proof inAppx. C.

Lemma 4.2. Any mean zero exclusion generator satisfies the strong sector condition (3.5) with aconstant C <∞ independent of ε > 0.

From (4.7), Lε − Tε satisfies the strong sector condition because we have matched drifts in (2.2). Inthe nearest neighbour case, i.e. ASEP, Mε = 0 and the strong sector condition is obvious.

Example 4.3. Suppose we add to our AEP generator Lε a generator Lpertε defined through a Dirichlet

form, i.e. −∫fLpertfdνε := Dpert

ε (f) which is comparable to Dε(f). One checks easily that theargument goes through. We have in mind Dirichlet forms of the following type,

ε−3/2∫ ∑

x∈εZαx(∇sym

x f)2 +∑

x,x′∈εZβx,x′(∇sym

x,x′f)2

dνε (4.8)

with functions αx, βx,x′ with αx not depending on the particle numbers at x−ε, x and βx,x′ not depend-ing on the particle numbers at x− ε, x, x′ − ε, x′, but on variables in a local (order ε) neighbourhoodof both, and∇sym

x,x′f(h) = f(hsymx,x′ )− f(h) where the height function hsymx,x′ is flipped both at x and x′,and the sum is over |x− x′| ≤ Cε. As long as αx, βx,x′ are uniformly bounded the Dirichlet form iscomparable to our standard one. The result is that Prop. 6.1 continues to hold. Of course, there aremany other examples of this type. The relevance of this example is that such models (i.e. with generatorLpert) are expected under appropriate scaling to have a hydrodynamic limit of Cahn-Hilliard type[2],[24]. Therefore they can be thought of as a very simple toy versions of the Kuramoto-Sivashinskyequation (though with a stabilizing instead of a destabilizing second order term) which has also beenconjectured to lie in the KPZ universality class [22].

5. DIRICHLET FORM OF THE TASEP TRANSITION PROBABILITIES

We will consider target sets B of the form ≤g for nice functions g where ≤g is the set

≤g= {h : h(x) ≤ g(x), x ∈ R} . (5.1)

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 7

Let pTASEP,εt (h,B) denote the probability for the rescaled TASEP height function at time t to be in set

B, given that initially it was h, and pAEP,εt (h,B) the analogue for AEP. We can follow the scheme of

Sec. 3. We will show in Appx. B that |∇symx pTASEP,ε

t (h,≤g)| is basically the joint probability

R(x, h(0); h) := PTASEP(

arg max{ht(y;−g) + h(y)} = x, ht(x;−g) + h(x) ∈ {ε1/2, 2ε1/2}).

(5.2)Here ht(·;−g) denotes the TASEP height function at time t started from −g. Except for the slightlyannoying issue of possible non-uniqueness of the argmax we would have that for each h, R(x, h(0); h)

is a probability measure on εZ× ε1/2Z. More precisely we will prove the following in Appx. B.

Lemma 5.1. For any g ∈ Sε such that g(x) ≥ C ′(1 + |x|) for some C ′ > 0 and any t > 0,∫ ( ∑x∈εZ

∑h(0)∈ε1/2Z

|∇symx pTASEP,ε

t (h,≤g)|)2µε(dh) ≤ C (5.3)

for some absolute constant C > 0, where µε is as defined in (2.5).

From this one might expect R(x, h(0); h) ∼ ε3/2. However, the Dirichlet form has the sums overεZ and ε1/2Z outside the square, and a computation gives Dε(pt) = O(1), from which (3.6) tells usnothing. The reason is that for fixed h, the argmax distribution is highly intermittent in x. We can getaround this by looking at slightly averaged transition probabilities.

We can have our shift operator τy use y ∈ R and act on functions on εZ by τyh(·) = h(· − [y]ε) andon the sets B by τyB = {h(·) : h(· − [y]ε) ∈ B} where [y]ε denotes the nearest point in εZ. Thereis another shift, σr which does the analogue to the heights themselves, σrh(x) = h(x) + [r]ε1/2 andσrB = {h(·) : h(·)− [r]ε1/2 ∈ B}. We will perform a little Gaussian average in r and y. Let y1, y2and r be independent, Gaussian, mean 0 and variance a2 for some a > 0 and let

pTASEP,εt (f0ν,B) := E[ pTASEP,ε

t (f0ν, σrτy1+y2B)] (5.4)

where E is the expectation over y1, y2, r. We could have combined y1 and y2 as y′ = y1 + y2 butwriting it this way will clarify the manipulations which follow. It is not hard to see that (3.1) becomes

pTASEP,εt (f0ν,B)− pAEP,ε

t (f0ν,B) =

∫ t

0

∫ft−s,ε(h)(Tε − Lε)pTASEP,ε

t (h,B)dνεds, (5.5)

pTASEP,εt (h,B) := E[ pTASEP,ε

t (τ−y1h, σrτy2B)]. (5.6)

Here we manipulated the last term a little bit, writing∫ft−s,ε(h)(Tε − Lε)pTASEP,ε

t (h, τy1+y2B)dνε =∫(Tε−Lε)∗ft−s,ε(h)pTASEP,ε

t (h, τy1+y2B)dνε then using pTASEP,εt (h, τy1+y2B) = pTASEP,ε

t (τ−y1h, τy2B).

Lemma 5.2. Let B be of the form (5.1). There exists a constant C > 0 such that for any fixed t > 0,with the definitions in the previous paragraph, and the Dirichlet form in (4.3),

Dε(pTASEP,εt (·,≤g)) ≤ Cε1/2a−2. (5.7)

Proof. The left hand side is given explicitly by

1

2ε−3/2

∑x∈εZ

∫Sε

(E[∇sym

x pTASEP,εs (τ−y1h, σrτy2B)]

)2dνε . (5.8)

Note here we apply the shifts on pTASEP,εs , then apply∇sym

x . Now using the notation that τy means τy,0,i.e. we only shift in the space variable, ∇sym

x τy = τy∇symx+y and by invariance of TASEP under spatial

and height shifts; pTASEP,εs (τyh, τyB) = pTASEP,ε

s (h,B) and pTASEP,εs (σrh, σrB) = pTASEP,ε

s (h,B). So∇symx (pTASEP,ε

s (τ−y1h, σrτy2B)) = (∇symx+y1p

TASEP,εs )(σ−rh, τy2+y1B). We can rewrite the expectation

over y1, y2 as the expectation in y′ = y2 + y1 of the expectation of y1 given y′. Now the distribution of

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 8

y1 given y′ is Gaussian, mean y′/2 and variance a2/2. Furthermore, by Jensen’s inequality, we cantake the expectation over y′ outside of the square. Putting this together, (5.8) is bounded above by

ε−5/2E

∫Sεε3/2

∑x∈εZ,h(0)∈ε1/2Z

(∫ϕ(y1, r)∇sym

x+y1pTASEP,εs (σrh, τy′B)dy1dr

)2

dµε

. (5.9)

where ϕ(y1, r) = e−(y1−y′/2)2

a2√πa2

e−r2

2a2√2πa2

and the outside expectation is over y′. By Young’s inequality

‖ϕ ∗ ψ‖2 ≤ ‖ϕ‖2‖ψ‖1 with ψ(x, r) = ∇symx pTASEP,ε

s (σrh, τy′B) we can bound

ε3/2∑x∈εZ

h(0)∈ε1/2Z

(∫ϕ(y1, r)∇sym

x+y1pTASEP,εs (σrh, τy′B)

)2

≤ Ca−2

ε3/2 ∑x∈εZ

h(0)∈ε1/2Z

∣∣∇symx pTASEP,ε

s (h, τy′B)∣∣

2

where the factor Ca−2 is the square of the L2 norm of ϕ. Together with (5.3) his gives (5.7). �

6. CONVERGENCE STARTING FROM FINITE ENERGY INITIAL DATA

Let pTASEP,εt (h,B), pAEP,ε

t (h,B) and pWASEP,εt (h,B) the transition probabilities for TASEP, AEP

and WASEP and pTASEP,εt (f0,ενε, B), pAEP,ε

t (f0,ενε, B) and pWASEP,εt (f0,ενε, B) denote the averaged

probabilities as in (5.4). From (3.6) and Lem. 5.2 we have∣∣∣pAEP,εt (f0,ενε, B)− pTASEP,ε

t (f0,ενε, B)∣∣∣ ≤ Ca−1‖f0,ε‖2 ε1/4 (6.1)

and ∣∣∣pWASEP,ε,δt (f0,ενε, B)− pTASEP,ε

t (f0,ενε, B)∣∣∣ ≤ Ca−1‖f0,ε‖2 δ1/2. (6.2)

If B is a set of the type (5.1) where g is uniformly continuous on all of R, the difference of τzB and Bcan be made arbitrarily small by choosing the range of averaging, a, small. Let us first draw conclusionsfrom (6.1). Since we know from [19] that pTASEP,ε

t (f0,εdνε,≤g)→ pFPt (ν,≤g), the KPZ fixed pointtransition probabilities, we thus have

Proposition 6.1. Suppose that f0,εdνε → dρ with ε1/4‖f0, ε‖2 → 0, and g is uniformly continuous onall of R with g(x) ≥ C(1 + |x|) for some C > 0. Then,

limε→0

pAEP,εt (f0,ενε,≤g) = pFPt (ρ,≤g). (6.3)

Remark 6.2. f0,εdνε → dρ with ‖f0,ε‖2 → 0, could, for example, follow closely some nice determin-istic function inside a box [−L,L] and then be Brownian motion outside, with f0,εdνε looking like anappropriate discretization. If the function g is not chosen to go to infinity quickly enough outside thebox, the right hand side of (6.3) may vanish, and the Proposition, while true, provides no information.

Remark 6.3. It is easy to see that everything above goes through if we replace the target set B by sets≥g where≥g= {h : h(x) ≥ g(x), x ∈ R} for any uniformly continuous g such that g(x) ≤ C(1−|x|)for some C > 0. The only change happens in the discussion preceding (5.3) where the arg max getsreplaced by arg min. Hence (5.3) still holds and we get the same statement as (6.3) with ≤g replacedby ≥g.

Next we draw conclusions from (6.2). Under TASEP, dh = −2ε−11∧dt + dM where M is amartingale, while under WASEP, dh = [2δε−3/2(1∨ − 1∧)− 2ε−11∧]dt+ dM where M is anothermartingale. Since

−21∧ = 12ε∇

−h∇+h+ 12ε

3/2∇−∇+h− 12 , 21∨ = −1

2ε∇−h∇+h+ 1

2ε3/2∇−∇+h+ 1

2 ,

where∇−h(x) = ε−1(h(x)− h(x− ε)),∇+h(x) = ε−1(h(x+ ε)− h(x)), and the martingales areapproximating white noises, one can see that these are discretizations of the KPZ equation. On the otherhand, it is also clear that for TASEP the second derivative term is too small and there is no way to scaleTASEP to the KPZ equation, while from the equation for WASEP one can read off the formal limit that

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 9

hWASEP(t, x) + 12ε−1t− t/4! converges as ε↘ 0 to the KPZ equation (1.1) with (λ, ν,D) = (12 , δ, δ).

This was proven in [3] under conditions which easily cover what we will use. On the other hand,in [19] it is shown that hTASEP(t, x) + 1

2ε−1t converges directly to the KPZ fixed point. The formal

argument above makes it clear why the normalization 12ε−1t is the same in both. Note that there is a

slight mismatch with the standard in [19] which sets λ = 14 instead of 1

2 . To keep things clean, in theproof we will use the 1

2 normalization and write KPZ′ and FP′ to indicate this change in factor, as wellas the t/4! shift. The conclusion is that as ε↘ 0, if f0,εdνε → dρ the left hand side of (6.2) convergesto |pKPZ’,δ

t (ρ,B)− pFP’t (ρ,B)| and therefore,

|pKPZ’,δt (ρ,B)− pFP’

t (ρ,B)| ≤ Cδ1/2a−1 limsupε→0

‖f0,ε‖2.

Again, if g is uniformly continuous the averaging makes an arbitrarily small error. So we have

Proposition 6.4. Suppose that f0,εdνε → dρ with ‖f0,ε‖2 ≤ C, and g is uniformly continuous on allof R with g(x) ≥ C(1 + |x|) for some C > 0. Then,

limδ→0

pKPZ,δt (dρ,≤g) = pFPt (dρ,≤g). (6.4)

Example 6.5. (Narrow wedges, randomized wedges and their approximations) For y ∈ R andb ∈ R the narrow wedge of height b at y for the KPZ fixed point is

dy,b(x) = b if x = y, dy,b(x) = −∞ otherwise.

For y1 < y2 < · · · < yk, b1, b2, . . . , bk ∈ R the multiple narrow wedge d~y,~b is d~y,~b(x) = maxi dyi,bi(x).It is useful to have some nice approximations which we call randomized wedges. For any k ∈

N, γ > 0, y1 < y2 < . . . < yk, b1, b2, . . . , bk ∈ R, define g(x) as the piecewise linear function suchthat g(x) = b1 for x ≤ y1, g(x) = bk for x ≥ yk and g(x), yi ≤ x ≤ yi+1 is the line segment joiningbi and bi+1 for i = 1, . . . , k − 1. Define the randomized wedge as

g(x) + Θ(x) ,

where Θ(x), x ∈ R is the following random process: Θ(yi) are uniformly distributed on [0, γ], i =1, . . . , k; Θ(y1 − x), x > 0, and Θ(yk + x), x > 0, are Brownian motions, and Θ(x), yi < x < yi+1

are Brownian bridges, all of them independent of each other. From their random walk approximationsin Sε, it is easy to see (see Appx. 6.5) that ‖f0,ε‖2 ≤ C independent of ε.

We give some special cases names and notations. The narrowish wedge of height b at x is dy,bs,L(x) =

max{b− s|x− y|,−L} and the multiple narrowish wedge is d~y,~b

s,L(x) = maxi dyi,bis,L (x). The Brownian

multiple narrowish wedge d~y,~b

γ,s,L is the randomized wedge approximating it. Note that the definition alsomakes sense when s and/or L is equal to∞. In Appx. 6.5 we will show that for any 0 < γ, s, L <∞,there exist f0,ε with

f0,εdνε → d~y,~b

γ,s,L ‖f0,ε‖L2(dνε) ≤ C. (6.5)

By skew time reversal invariance of KPZ and ASEP and (6.5) we obtain,

Proposition 6.6. Suppose that h(x) ≤ A(1− |x|) for some A and h is uniformly continuous. Then forany Brownian multiple narrowish wedge or randomized wedge (see Defn. 6.5),

limδ→0

pKPZ,δt (h,≤

−d~y,~b

γ,s,L

) = limε→0

pASEP,εt (h,≤

−d~y,~b

γ,s,L

) = pFPt (h,≤−d~y,

~bγ,s,L

).

The rest of the work of the paper is to use properties of AEP, ASEP and the KPZ equation to extendthe result to distributional convergence in the uniform-on-compact topology, and broaden the class ofinitial data.

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 10

7. TIGHTNESS AND CONVERGENCE IN THE UNIFORM-ON-COMPACT TOPOLOGY

Proposition 7.1. Let t > 0 be fixed and let hε(·) be some height function in Sε and assume that for allg1, g2 (uniformly) continuous with g1(x) = C1 + C2|x− c3| and g2(x) = C1 − C2|x− c3| for all xlarge enough, for some C1, C2 > 0 and c3 ∈ R, we have

limε→0

P(hε(x) ≤ g1(x) ∀x ∈ R) = pFPt (h,≤ g1), limε→0

P(hε(x) > g2(x)∀x ∈ R) = pFPt (h,≥ g2) ,(7.1)

for some function h which is bounded on any compact interval and |h(x)| ≤ C(1 + |x|) for someC > 0. Then {hε}ε>0 is tight in the uniform-on-compact topology. Moreover, we have the followingmodulus of continuity uniform in ε. For any b > 0 small enough,

limsupε→0

P(ωε(b) > mb1/2 log7/6(1 + b−1)) ≤ ce−dm3/2,

where ωε(r) := sup|u−v|≤r,u,v∈[−1,1] |hε(u)− hε(v)| for 0 < r < 1.

Since ≤g sets for uniformly continuous g growing to∞ at some rate form a separating class for theKPZ fixed point started from any h with h(x) ≤ C(1 + |x|), by Prop. 6.1, Rem. 6.3 and Prop. 7.1 weget the uniform-on-compacts part of Thm. 2.1(1).

Proof. Without loss of generality we can assume that the compact interval is [−1, 1]; so it is enough toshow that {hε|[−1,1]}ε is tight in the uniform topology. Throughout c, d will denote positive universalconstants whose values can change from line to line.

For any e ∈ [−1, 1] define for all x ∈ R,

se(x) = (|x− e|1/2 log1/2(1 + |x− e|−1) ∧ 1) + |x− e| .

Let hFPt denote the KPZ fixed point at time t started from the initial condition h. Then there exists arandom constant K0 > 0 such that for all e ∈ [−1, 1], |hFPt (e)| ≤ K0, where K0 satisfies P (K0 >

m) ≤ ce−dm3/2

for m > 0. Also since |h(x)| ≤ C(1 + |x|) for some C > 0, hence there exists arandom constant K1 > 0 such that for all e ∈ [−1, 1], and all x /∈ [−2, 2],

|hFPt (x)− hFPt (e)| ≤ K1|x− e| ≤ K1se(x) ,

where K1 satisfies P (K1 > m) ≤ ce−dm3/2for m > 0. For all e ∈ [−1, 1] and x ∈ [−2, 2],

|hFPt (x)− hFPt (e)| = | supz∈[−K2,K2]

(h(z) +A(z, x))− supz∈[−K2,K2]

(h(z) +A(z, e))|

≤ supz∈[−K2,K2]

|A(z, x)−A(z, e)| , (7.2)

where A(·, ·) denotes the Airy sheet and K2 is a random constant such that P (K2 > m) ≤ ce−dm3.

Using Proposition 10.5 of [9] and putting all this together, we thus have for all e ∈ [−1, 1], and allx ∈ R,

|hFPt (x)− hFPt (e)| ≤ Kse(x) , and |hFPt (e)| ≤ K , (7.3)

for some random constant K that satisfies P (K > m) ≤ ce−dm3/2for m > 0. We call the above event

EK .Now fix any k > 0 large and a, b > 0 small (to avoid cumbersome notations we assume k, a−1, b−1 ∈

N). Now we consider the rectangle [−1, 1]× [−k, k] and split it into grids with rectangles of vertical andhorizontal dimensions a and b respectively. That is, let ei = −1 + ib for i ∈ I := {0, 1, 2, . . . , 2b−1}and fj = −k + ja for j ∈ J := {0, 1, . . . , 2ka−1}. Then by (7.1), for any i, j, P(hε(x) >fj+1 + ksei(x) for somex ∈ R) → P(hFPt (x) > fj+1 + ksei(x) for somex ∈ R) and P(hε(x) ≤fj − ksei(x) for somex ∈ R)→ P(hFPt (x) ≤ fj − ksei(x) for somex ∈ R) as ε→ 0. Since on theevent Ek,

{hFPt (x) > fj+1 + ksei(x) for somex ∈ R} ⊆ {hFPt (ei) > fj+1} and

{hFPt (x) ≤ fj − ksei(x) for somex ∈ R} ⊆ {hFPt (ei) ≤ fj} ,

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 11

we have, for Ti,j,k := {(x, y) ∈ R2 : fj − ksei(x) < y ≤ fj+1 + ksei(x)} denoting the (i, j)-th tube,using a union bound,

limsupε

P((x, hε(x)) ∈ T ci,j,k for somex ∈ R) ≤ P(hFPt (ei) ∈ (fj , fj+1]c) + 2P(Eck) .

That is, liminfε P(hε ∈ Ti,j,k) ≥ P(hFPt (ei) ∈ (fj , fj+1])− 2P(Eck). (By an abuse of notation, hε herealso denotes the graph (x, hε(x)) for all x ∈ R.) For a fixed i, since the sets Ti,j,k are disjoint for allj = 0, 1, 2, . . . , 2ka−1, summing the above probabilities, we have again by (7.3),

liminfε

P(hε ∈ Ti,j,k for some j) ≥ P(hFPt (ei) ∈ [−k, k])−(4ka−1+1)P(Eck) ≥ 1−(4ka−1+2)P(Eck) .

Taking a union bound over all i = 0, 1, . . . , 2b−1,

liminfε

P(Gεa,b,k) ≥ 1− 10ka−1b−1P(Eck) , (7.4)

where the event Gεa,b,k is defined as

Gεa,b,k := {∀i ∈ I , hε ∈ Ti,j,k for some j ∈ J } .

Now let ωε(r) := sup|u−v|≤r,u,v∈[−1,1] |hε(u) − hε(v)| for 0 < r < 1. For any u, v ∈ [−1, 1] with|u − v| ≤ b, there exists some i ∈ I such that |u − ei| ≤ b and |v − ei| ≤ b. Then on the eventGεa,b,k, if j ∈ J is such that hε(ei) ∈ (fj , fj+1], we have hε ∈ Ti,j,k. Hence for b small enough,|hε(u)−hε(ei)| ≤ a+ksei(u) ≤ a+ 2kb1/2 log1/2(1 + b−1), and the same holds for |hε(v)−hε(ei)|.Thus

ωε(b) ≤ 2a+ 4kb1/2 log1/2(1 + b−1) .

Choosing a = b1/2, we have for b small enough, by (7.4) and (7.3),

limsupε

P(ωε(b) > kb1/2 log1/2(1 + b−1)) ≤ 60kb−3/2P(Eck) ≤ ckb−3/2e−dk3/2 ≤ c′b−3/2e−d′k3/2 .

Choosing k = Cm log2/3(1 + b−1), we have

limsupε

P(ωε(b) > mb1/2 log7/6(1 + b−1)) ≤ ce−dm3/2.

Now fix any η > 0, λ > 0. Choose m large enough so that ce−dm3/2

< η. Now choose b smallenough so that mb1/2 log7/6(1 + b−1) < λ. Setting δ = b, we have limsupε P(ωε(δ) > λ) < η.Also by letting one of the ei to equal 0, it is easy to see from above that there exists k > 0 such thatlimsupε P(|hε(0)| > k) < η. Hence {hε|[−1,1]}ε is tight in the uniform topology. �

8. DETERMINISTIC INITIAL CONDITIONS FOR ASEP

In this section, we use the skew-time reversibility/order-preserving property of ASEP height functionto extend our result in (6.3) to a broader class of initial conditions. Although the results are statedfor ASEP, the same proofs work for the KPZ equation, which inherits these properties in the weaklyasymmetric limit. They do not work for finite range exclusions, AEP, because the order of heightfunctions is not preserved. The fact that in reverse time, the height function evolves as minus theoriginal height function, is common to all these models. However, the statement (1.4) requires thepreservation of order as well.

8.1. Initial conditions decaying to −∞ with target functions going to ∞. Let h be uniformlycontinuous and g be continuous with h(x) ≤ C1(1−|x|) and g(x) ≥ C2(1 + |x|) for some C1, C2 > 0and for all x. We will prove now that

limε→0

pASEP,εt (h,≤g) = pFPt (h,≤g). (8.1)

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 12

Here and henceforth the initial condition h for ASEP at scale ε is to be interpreted as a discreteapproximation of h in Sε. Using skew time reversibility and Prop. 6.6, we have for any randomizedwedge ρ, and any h as above,

limε→0

pASEP,εt (h,≤ρ) = pFPt (h,≤ρ). (8.2)

Starting from h, with probability one, for some random positive constant C0, one has the KPZ fixedpoint hFPt (x) ≤ C0(1− |x|) for all x. We claim that given any η > 0, there exists a randomized wedgeρ such that

liminfε→0

pASEP,εt (h,≤ρ) ≥ 1− η . (8.3)

Indeed, by choosing the randomized wedge ρ such that pFPt (h,≤ρ) ≥ 1 − η, we have by (8.2) thatliminfε→0 p

ASEP,εt (h,≤ρ) = pFPt (h,≤ρ) ≥ 1− η.

Now for any g as in (8.1), and any large enough M > 0, construct two randomized wedges ρ1, ρ2such that ρ1 ≤ g with probability 1− η; ρ2|[−M,M ] ≥ g|[−M,M ] and ρ2|[−M,M ]c ≥ ρ|[−M,M ]c for somerandomized wedge ρ that satisfies (8.3) with probability 1− η. Then by (8.2)

pFPt (h,≤ρ1)− η ≤ liminfε→0

pASEP,εt (h,≤g) .

Moreover, for any fixed t > 0, if hε = hε(t, ·) denotes the ASEP height function at time t started fromh (defined in Theorem 2.1), then on the event that ρ2|[−M,M ] ≥ g|[−M,M ] and ρ2|[−M,M ]c ≥ ρ|[−M,M ]c ,we have

{hε ≤ g} ⊆ {hε(x) ≤ g(x)∀x ∈ [−M,M ]} ⊆ {hε(x) ≤ ρ2(x)∀x ∈ [−M,M ]}⊆ {hε ≤ ρ2} ∪ {hε(x) > ρ(x) for some x ∈ [−M,M ]c} .

Hence from (8.3) and (8.2),

limsupε→0

pASEP,εt (h,≤g) ≤ η+limsup

ε→0pASEP,εt (h,≤ρ2)+limsup

ε→0(1−pASEP,ε

t (h,≤ρ)) ≤ pFPt (h,≤ρ2)+2η .

Also if M is large enough so that the KPZ fixed point hFPt (x) started from h at time t is less thanρ1(x), ρ2(x) and g(x) for all x ∈ [−M,M ]c with probability 1− η; and ρ1|[−M,M ] and ρ2|[−M,M ] areclose to g|[−M,M ] such that by the continuity of the KPZ fixed point hFPt , the probabilities of the events{hFPt (x) ≤ ρ2(x) ∀x ∈ [−M,M ]}, {hFPt (x) ≤ g(x) ∀x ∈ [−M,M ])}, and {hFPt (x) ≤ ρ1(x)∀x ∈[−M,M ]} are η-close to each other, then from the last equation we have

pFPt (h,≤g)− 4η ≤ liminfε→0

pASEP,εt (h,≤g) ≤ limsup

ε→0pASEP,εt (h,≤g) ≤ pFPt (h,≤g) + 5η .

Since η is arbitrary, we have the result.The same proof works for the KPZ equation, we record the result here.

Remark 8.1. Let h be uniformly continuous and g be continuous with h(x) ≤ C1(1 − |x|) andg(x) ≥ C2(1 + |x|) for some C1, C2 > 0, for all x large enough. Then

limδ→0

pKPZ,δt (h,≤g) = pFPt (h,≤g). (8.4)

8.2. General initial conditions with target functions going to∞. Let h be a continuous functionsuch that h(x) ≤ C1(1 + |x|1/2) for some C1 > 0, and g be uniformly continuous with g(x) ≥C2(1 + |x|) for some C2 > 0. We will prove in this subsection that

limε→0

pASEP,εt (h,≤g) = pFPt (h,≤g) . (8.5)

By skew time reversibility, it is equivalent to prove the convergence for h uniformly continuous withh(x) ≤ C1(1− |x|) for some C1 > 0 and g continuous with g(x) ≥ C2(1− |x|1/2).

First we claim that for any η > 0 and any randomized wedge ρ such that ρ(0) is bounded, thereexists M > 0 such that liminfε→0 p

ASEP,εt (h,≤ρ|[−M,M ]c

) > 1−η (where for any A ⊆ R, ρ|A denotesthe function on R ∪ {∞} which is equal to ρ on A and∞ on Ac). Indeed, by choosing M > 0 large

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 13

enough and a randomized wedge ρ′ such that ρ′|[−M,M ]c = ρ|[−M,M ]c and pFPt (h,≤ρ′) > 1− η, onehas

liminfε→0

pASEP,εt (h,≤ρ|[−M,M ]c

) ≥ liminfε→0

pASEP,εt (h,≤ρ′) = pFPt (h,≤ρ′) > 1− η .

Now let ρ(x) = U + Θ(x), where Θ is a standard two-sided Brownian motion and U is uniform in[C2− 1, C2] and Θ and U are independent. Then ρ is a randomized wedge and thus there exists M > 0

(that depends only on η and C2) such that liminfε→0 pASEP,εt (h,≤ρ|[−M,M ]c

) > 1− η, that is,

limsupε→0

P(hε(x) ≥ ρ(x) for some x ∈ [−M,M ]c) < η . (8.6)

where hε(·) = hε(t, ·) denotes the ASEP height function at time t started from h for any fixed t > 0(see Theorem 2.1). We claim that

limsupε→0

P(hε(x) ≥ g(x) for some x ∈ [−M,M ]c) ≤ βη (8.7)

where β > 0 is some constant depending only on C2. Indeed let

β−1 := P(Θ(1) < −C2) = P(Θ(x) < −C2|x|1/2)

for any x ∈ R. Then for any x ∈ R, P(ρ(x) < g(x)) ≥ β−1. Because of the independence of hε and ρ,

P(hε(x) ≥ ρ(x) for some x ∈ [−M,M ]c) ≥ P(hε(x) ≥ g(x), g(x) > ρ(x) for some x ∈ [−M,M ]c)

≥ β−1P(hε(x) ≥ g(x) for some x ∈ [−M,M ]c) .

Thus (8.7) follows from (8.6).Next let g1 be a continuous function such that g1(x) ≥ C(1 + |x|) for some C > 0 and g1|[−M,M ] =

g|[−M,M ] and g1(x) ≥ g(x) for all x. Then pASEP,εt (h,≤g) ≤ pASEP,ε

t (h,≤g1). Also, sinceg1|[−M,M ] = g|[−M,M ], we have

{hε ≤ g1} ⊆ {hε(x) ≤ g1(x)∀x ∈ [−M,M ]} = {hε(x) ≤ g(x)∀x ∈ [−M,M ]}⊆ {hε ≤ g} ∪ {hε(x) > g(x) for some x ∈ [−M,M ]c} .

Thus by (8.7),liminfε→0

pASEP,εt (h,≤g) ≥ liminf

ε→0pASEP,εt (h,≤g1)− βη .

By (8.1) of the last subsection and the growth condition on g1, limε→0 pASEP,εt (h,≤g1) = pFPt (h,≤g1).

Also if M is large enough so that the KPZ fixed point hFPt (x) started from h at time t is less than g(x)and g1(x) for all x ∈ [−M,M ]c with probability 1− η, we have

pFPt (h,≤g)− (β + 2)η ≤ liminfε→0

pASEP,εt (h,≤g) ≤ limsup

ε→0pASEP,εt (h,≤g) ≤ pFPt (h,≤g) + 2η .

Since η is arbitrary, we have the result.

Remark 8.2. Using similar argument as above, starting from Rem. 6.3 we will likewise have thefollowing result. Let h be a continuous function such that C0(1 + |x|) ≥ h(x) ≥ C1(1− |x|1/2) forsome C0, C1 > 0, and g be uniformly continuous with C2(1 − |x|) ≤ g(x) ≤ C3(1 − |x|) for someC2, C3 > 0, for all x large enough. Then limε→0 p

ASEP,εt (h,≥g) = pFPt (h,≥g).

Observe that Subsec. (8.2) and Rem 8.2, together with Prop. 7.1 in the previous section, give thetightness of the ASEP height function started from any h continuous such that |h(x)| ≤ C(1 + |x|1/2)for some C > 0 and hence shows the convergence in the uniform-on-compact topology. This provesthe first statement in Thm. 2.1 (2).

Now from the convergence in the uniform-on-compact topology we have convergence of finitedimensional distributions, i.e. for any h continuous such that for some C > 0, |h(x)| ≤ C(1 + |x|1/2),

limε→0

pASEP,εt (h,≤−d~y,~b) = pFPt (h,≤−d~y,~b). (8.8)

where d~y,~b is the multiple narrow wedge defined in Def. 6.5.

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 14

By the definition of ASEP height function, it is easy to see that hε(t, y) ≤ b for some y, b ∈ R isequivalent to hε(t, y + εz) ≤ b + ε1/2|z| for all z ∈ R. Thus if we define for y, b ∈ R, dy,bε (x) =

b − ε−1/2|x − y|, and for y1 < y2 < · · · < yk, b1, b2, . . . , bk ∈ R, d~y,~b

ε (x) = maxi dyi,biε (x), then

pASEP,εt (h,≤−d~y,~b) = pASEP,ε

t (h,≤−d~y,

~bε

). Hence, applying skew time reversibility to (8.8), we havethe convergence for multi-narrow wedge initial condition. That is, for any continuous h such that forsome C > 0, |h(x)| ≤ C(1 + |x|1/2),

limε→0

pASEP,εt (d~y,

~bε ,≤h) = pFPt (d~y,

~b,≤h). (8.9)

This proves the third statement in Thm. 2.1 (2).On the other hand, starting with these wedges, we don’t have the tightness. For a single narrow

wedge, Theorem 1.1 of [6] gives the tightness of the ASEP height function. Thus (8.9) gives theconvergence of the single narrow wedge ASEP height function to the Airy2 process (after subtractingthe required parabola) in the uniform on compact topology for a fixed t > 0. This proves the secondstatement in Thm. 2.1 (2).

9. BOOTSTRAPPING KPZ USING UNIFORM HÖLDER CONTINUITY

Though the results in Sec. 7 and Sec. 8 also work for the KPZ equation, the KPZ equation enjoysadditional properties that allows us to extend our results in the previous sections to an even broaderclass of initial conditions.

The KPZ,δ proto-Airy sheet Aδ,t(x, y) is defined as follows. First of all we consider the white noisein KPZ,δ (2.8) to be fixed and solve with any admissible data using that same white noise, to produce astochastic flow on the space of admissible data. Note that the solution of KPZ,δ (2.8) with nice initialdata h(0, x) = h0(x) is simply defined as h(t, x) = 4δ logZ(t, x) where Z(t, x) solved the stochasticheat equation

∂tZ = δ∂2xZ + 14δ

1/2ξZ, (9.1)

with initial data Z(0, x) = e(4δ)−1h0(x). The "initial data" narrow wedge at y for KPZ,δ(2.8) means

h(t, x) = 4δ logZ(t, x) where Z(0, x) = δy(x). We call the process with the parabola removedh(t, x) + 1

t (x− y)2 := Aδ,t(x, y). By linearity in the initial data of the stochastic heat equation (9.1)we have for any nice initial data h0(x),

h(t, y;h0) = 4δ log∫

exp{

(4δ)−1(Aδ,t(x, y)− 1

t (x− y)2 + h0(x))}dx. (9.2)

where h(t, y;h0) is the solution of KPZ,δ starting from h0.

Let ‖ · ‖α,B denote the Hölder norm ‖f‖α,B = supx1,x2|f(x2)−f(x1)||x2−x1|α where the supremum is over

[−B,B] or [−B,B]2, depending on whether the function has one or two real variables.

Lemma 9.1. [7]

(1) For any 0 < α < 1/2 and B < ∞, limK→∞ P(‖Aδ,t(x, y)‖α,B ≥ K) = 0 uniformly inδ > 0;

(2) For any b > 0, limA→∞ P(Aδ,t(x, y) ≥ A+ b(|x|2 + |y|2)) = 0 uniformly in δ > 0.

Proof. Part (1) follows from the absolute continuity of the narrow wedge KPZ equation with respect toBrownian motion on any compact set and the tightness of the Radon-Nikodym derivative for all δ > 0(see Theorem 1.2 of [7]) and the symmetry in the two coordinates of Aδ,t(x, y).

Part (2) follows from Lemma 4.1 of [7] and the symmetry in the two coordinates of Aδ,t(x, y). �

Suppose our initial data h0 satisfies h0(x) ≤ A + b|x|2 for all x ∈ R, for some b < (2t)−1. Letσ > 0. Then, with probability greater than 1− σ, we can choose B large enough that the integral in(9.2) over |x| > B is bounded by σ, uniformly in δ > 0. It is also immediate from (9.2) that for any0 < α < 1/2 and B < ∞, limK→∞ P(‖h(t, ·)‖α,B ≥ K) = 0 uniformly in δ > 0. This gives thetightness of h(t, ·).

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 15

Suppose in addition that h0 is uniformly continuous with h0(x) ≤ A(1− |x|) for some large enoughA. Let σ > 0 and b < t−1. IfA is large enough then the probability thatAδ,t(x, y) ≤ A+b(|x|2+ |y|2)is larger than 1 − σ. It is not hard to see from (9.2) that then there is an L0 and a ρ > 0 so thath(t, x) ≤ L0 − ρ|x|, for all x and δ with probability greater than 1 − σ. Let g be any continuousfunction such that g(x) ≤ C(1 + |x|) for all x for some 0 < C < ρ. Then there is an M > 0 such thatfor all δ and all x such that |x| > M , h(t, x) ≤ −g(x) with probability greater than 1− σ. Construct acontinuous function g1 such that g1(x) = −g(x) for all x ∈ [−M,M ], g1(x) ≥ −g(x) for all x andg1(x) > C1(1 + |x|) for some C1 > 0 and all large x. Then

pKPZ,δt (h0,≤g1)− σ ≤ pKPZ,δ

t (h0,≤−g) ≤ pKPZ,δt (h0,≤g1) .

By (8.4), limδ→0 pKPZ,δt (h0,≤g1) = pFPt (h0,≤g1). Also if M is large enough so that the KPZ fixed

point started from h0 at time t is less that −g(x) and g1(x) for all x ∈ [−M,M ]c with probability1− σ, we get, since σ can be made arbitrarily small,

limδ→0

pKPZ,δt (h0,≤−g) = pFPt (h0,≤−g). (9.3)

From (9.3), using skew time reversibility, we get for any continuous g such that for some C > 0,g(x) ≤ C(1 + |x|) for all x and any uniformly continuous h0 such that h0(x) ≥ A(1 + |x|) for somelarge enough A,

limδ→0

pKPZ,δt (g,≤h0) = pFPt (g,≤h0). (9.4)

This, together with the tightness of h(t, ·) gives the convergence of the KPZ equation started from g tothe KPZ fixed point in distribution in the topology of uniform-on-compacts.

Next we work towards multi-narrow wedge initial data where Z(0, x) = d~x,~a(x). An approximationof it in the α-Hölder class satisfying h0(x) ≤ A − |x| is d

~x,~a+(δ/2) log(δ/2)−1sL,s,A where d~x,~aL,s,A(x) =

min{d~x,~aL,s(x), A − |x|}. Let σ > 0. On a set with probability greater than 1 − σ, the α-Höldernorm in y of Aδ,t(x, y), on set [x − s−1(a + L), x + s−1(a + L)] × [−B,B] is less than K andAδ,t(x, y) ≤ A+ b(|x|2 + |y|2) for some b < t−1. Here x, x are the smallest and largest xi and a, aare their corresponding ai. The second interval is just the smallest interval containing the wedge parts

of d~x,~a+δ log 1

2 δ−1s

L,s,A . Since these wedge parts can vary at most rs,δ,L := maxi s−1(ai + δ log δ−1s+ L),we have, on this set,

4δ log(∑

µie(4δ)−1(ai+Aδ,t(xi,y)−

1t (xi−y)

2−Krαs,δ,L) +RL(y))≤ h(t, y; d

~x,~a+(δ/2) log(δ/2)−1sL,s,A )

≤ 4δ log(∑

µie(4δ)−1(ai+Aδ,t(xi,y)−

1t (xi−y)

2+Krαs,δ,L) +RL(y)),

where µi = 12(4δ)−1s

∫ s−1(ai+L)−s−1(ai+L)

e−(4δ)−1s|x|dx and

RL(y) :=

∫exp

{(4δ)−1

(Aδ,t(x, y)− 1

t (x− y)2 + (−L) ∨ (A− |x|))}dx.

Since Aδ,t(x, y) ≤ A+ b(|x|2 + |y|2), RL → 0 uniformly on compact sets of y as L→∞. Lettingδ →∞ followed by L, s→∞ we have

limsupδ→0

∣∣P(maxi,j{ai + bj +Aδ,t(xi, yj)− 1

t (xi − yj)2} ≤ 0)− pFPt (d~x,~a,≤−d~y,~b)

∣∣ ≤ σ.Since σ > 0 is arbitrary, we have proved the convergence of the finite dimensional distributions of theKPZ,δ starting with k narrow wedges to the KPZ fixed point. Together with the tightness of the KPZ,δheight function, we get the distributional convergence in the uniform-on-compact topology. A similarargument proves the same result starting with k narrow wedges plus a continuous function boundedabove by some A(1 + |x|). This proves the first statement in Thm. 2.1 (3).

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 16

10. CONVERGENCE OF THE KPZ LINE ENSEMBLE TO THE AIRY LINE ENSEMBLE

Recall the KPZt ensemble from [7]. The Hamiltonian Ht : R 7→ [0,∞) is Ht(x) = et1/3x. Let L =

(L1,L2, . . .) be an N× R indexed line ensemble. Fix k1 ≤ k2 with k1, k2 ∈ Z, an interval (a, b) ⊂ Rand two vectors ~x, ~y ∈ Rk2−k1+1. Given two measurable functions f, g : (a, b) 7→ R ∪ {±∞}, thelaw Pk1,k2,(a,b),~x,~y,f,gHt

on Lk1 , . . . ,Lk2 : (a, b) 7→ R has the following Radon-Nikodym derivative with

respect to Pk1,k2,(a,b),~x,~yfree , the law of k2 − k1 + 1 independent Brownian bridges taking values ~x at timea and ~y at time b:

Pk1,k2,(a,b),~x,~y,f,gHt

Pk1,k2,(a,b),~x,~yfree

(Lk1 , . . . ,Lk2) =exp

{−∑k2

i=k1−1∫ ba Ht(Li+1(u)− Li(u))du

}Zk1,k2,(a,b),~x,~y,f,gHt

,

with Lk1−1 = f and Lk2+1 = g, and Zk1,k2,(a,b),~x,~y,f,gHtis the normalizing constant. We say that the

line ensemble L has the Ht-Brownian Gibbs property if for all K = {k1, . . . , k2} ⊂ N and (a, b) ⊂ R,the conditional distribution of L|K×(a,b) given L|(N×R)\(K×(a,b)) is Pk1,k2,(a,b),~x,~y,f,gHt

. Here f = Lk1−1and g = Lk2+1, with the convention that if k1 = 1 then f ≡ +∞.

Theorem 10.1 ([7], Thm. 2.15). For all t ≥ 1, there exists an N × R indexed line ensemble ht =(ht1, h

t2, . . .) such that

(1) The lowest indexed curve (top line) ht1 : R 7→ R is equal in distribution to the scaled time tHopf Cole solution to the narrow wedge initial data KPZ equation.

(2) The ensemble ht has the Ht-Brownian Gibbs property.

We call any such line ensemble a (scaled) KPZt line ensemble.

Building on our work in the last section, we are now ready to prove the convergence4 of the entire(scaled) KPZt line ensemble to the Airy line ensemble, thereby proving Conjecture 2.17 of [7]. Itfollows from our main result combined with the following two recent results:

Theorem 10.2 ([31]). For t > 0 the scaled KPZt line ensemble is tight and any subsequential limit is anon-intersecting line ensemble with the Brownian Gibbs property.

Theorem 10.3 ([10]). A Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional distributions of its top curve.

Corollary 10.4. The N× R indexed line ensemble defined by the map (n, x) 7→ 21/3(htn(x) + x2/2)converges in distribution as a line ensemble to the Airy line ensemble {An(x) : n ∈ N, x ∈ R} ast→∞, in the uniform-on-compact topology.

Proof. By Thm. 10.2 and Prohorov’s theorem, it is enough to show that any subsequential limit of thescaled KPZt line ensemble is the Airy line ensemble (after the parabolic shift). To this end, observe thatby our main result the finite dimensional distributions of the top line of the scaled KPZt line ensemble,which is the the scaled time t Hopf Cole solution to the narrow wedge initial data KPZ equation,converge to those of the top line of the Airy line ensemble. Hence the finite dimensional distributionsof the top line of any subsequential limit of the scaled KPZt line ensemble match with those of the topline of the Airy line ensemble. Since any subsequential limit also has the Brownian Gibbs property, itfollows from Thm. 10.3 that it is the Airy line ensemble (after the parabolic shift). �

APPENDIX A. PROOF OF (6.5)

Fix any 0 < γ, s, L <∞. Approximate ν by the corresponding random walk-approximation in theSε-lattice and by replacing yi by [yi]ε, bi by [bi]ε1/2 , γ by [γ]ε1/2 , L by [L]ε1/2 (where [x]ε denotes thenearest point in εZ).

4Here we are taking t→∞. To harmonize with the previous section take t = δ−3.

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 17

First we show that if ηaε denotes the random walk approximation of a standard Brownian mo-tion with drift a on [0, d], then the Radon Nikodym derivative fε of ηaε with respect to η0e satisfieslimsupε

∫f2ε dη

0ε < ∞. (By an abuse of notation, ηaε denotes both a measure and its realization as a

random walk.) Recall that ηaε is constructed as follows. Let B be a standard Brownian motion. For anyx ∈ {0, 1, . . . , [dε−1]}, if B((x + 1)ε) − B(xε) + aε > 0 then ηaε ((x + 1)ε) − ηaε (xε) = ε1/2, elseηaε ((x+ 1)ε)− ηaε (xε) = −ε1/2. Then for any sequence ξ of length n = [dε−1] with entries in ±ε1/2,fε(ξ) = 2npSnn (1 − pn)n−Sn , where pn = P (B(ε) + aε > 0) and Sn denotes the number of ε1/2’s

in ξ. Thus (here E denotes expectation with respect to η0ε ), Ef2ε = 22n(1 − pn)2nE(

pn1−pn

)2Sn=

(1 + (2pn− 1)2)n ≤ en(2pn−1)2 . Since√n(pn− 1/2) =

√n(P (B(1) > −a

√ε)− 1/2)→ a

√dφ(0),

where φ is the density of a standard Gaussian variable,

limsupε

∫f2ε dη

0e ≤ e4a

2dφ2(0) . (A.1)

Now we prove (6.5). First, for simplicity, assume that k = 1. Due to translation invariance of νε, wecan assume, without loss of generality, that y1 = 0. Then for any curve ρ in Sε, by conditioning onρ(0) ∈ {[b1]ε1/2 , . . . , [b1]ε1/2 + [γ]ε1/2} and using (A.1) on [−s−1(ρ(0) + L), s−1(ρ(0) + L)],

limsupε

Vε = limsupε

∫f0,ε(ρ)2dνε ≤ 2γ−1 exp

{4s(b1 + L+ γ)φ2(0)

}. (A.2)

Next we prove the proposition for k = 2; it is easy to see that the general case is conceptually similarto this with heavier notations. Again we can assume y1 = 0. Let µ′ε, η

0ε denote the restrictions of f0,ενε

and νε on [0, y2] with ρ(0) ≡ 0 (that is, η0ε is the random walk measure starting from 0). Because of(A.2) and conditioning on ρ(0), it is enough to prove that the Radon-Nikodym derivative gε of µ′ε with

respect to η0ε satisfies limsupε∫g2εdη

0ε <∞. Let ψ(x) = d~y,

~bs,L(x) with k = 2 and y1 = 0, b1 = 0. Let

ηψε denote the random walk approximation of B(x) + ψ(x) on [0, y2], where B is a standard Brownianmotion. Then for any curve ρ in Sε with ρ(0) = 0 and ρ(y2) ∈ {[b2]ε1/2 , . . . , [b2]ε1/2 + [γ]ε1/2},

gε(ρ) =µ′ε(ρ)

η0ε(ρ)≤ ηψε (ρ)P(U = ρ(y2)− [b2]ε1/2)

P(η0ε(y2) = ρ(y2)− [b2]ε1/2)η0ε(ρ)≤ Cγ,y2

ηψε (ρ)

η0ε(ρ), (A.3)

where U denotes a discrete uniform random variable on {0, 1, 2, . . . , [γ]ε1/2} andCγ,y2 is some constantdepending only on y2, γ. Following the same argument as in (A.1), the Radon-Nikodym derivative ofηψε with respect to η0ε is bounded in L2(η0ε). Hence, from (A.3), it follows that limsupε

∫g2εdη

0e <∞.

APPENDIX B. PROOF OF LEM. 5.1

Lemma B.1. Let X(ht, h) denote the number of points of maxima of ht(·;−g) + h(·). Then∑x∈εZ

∑h(0)∈ε1/2Z

|∇symx pTASEP,ε

t (h,≤g)| ≤∞∑k=1

kPTASEP(X(ht, h) = k) = ETASEP(X(ht, h)) . (B.1)

Proof. For simplicity of notation, we denote hx,1,sym by hx. First of all if hx = h, then clearly∇symx pTASEP,ε

t = 0. On the other hand, if h has a local maximum at x, then, by skew-time reversibilityof TASEP height function (see (1.4)), if ht(y;−g) = hεt (y;−g) denotes the TASEP height function atscale ε at time t and position y started from −g, |∇sym

x pTASEP,εt (h,≤g)| is given by∣∣∣pTASEP,ε

t (hx,≤g)− pTASEP,εt (h,≤g)

∣∣∣ = pTASEP,εt (−g,≤−hx)− pTASEP,ε

t (−g,≤−h),

which is the probability that ht(y;−g) ≤ −hx(y) for all y ∈ εZ, but for some y, ht(y;−g) > −h(y).Of course, necessarily y = x, and one can see that this is the probability that ht(y;−g) + h(y) ≤ 0 forall y and ht(x;−g) + h(x) is ε1/2 or 2ε1/2. Thus

|∇symx pTASEP,ε

t (h,≤g)| ≤ PTASEP(

arg max{ht(y;−g)+h(y)} = x, ht(x;−g)+h(x) ∈ {ε1/2, 2ε1/2}).

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 18

Here PTASEP denotes the probability with respect to the TASEP dynamics for a fixed h. The right-hand-side above is equal to PTASEP(arg max{ht(y;−g) + h(y)} = x, ht(x;−g) + h(x) ∈ {ε1/2 −h(0), 2ε1/2 − h(0)}), so this probability summed over all h(0) ∈ ε1/2Z for each fixed h equalsPTASEP(arg max{ht(y;−g) + h(y)} = x). Note that only one of the two possibilities ht(x;−g) +

h(x) = ε1/2, ht(x;−g) + h(x) = 2ε1/2 happens; it is because there is a parity conservation in themodel.

On the other hand, if h has a local minimum at x, |∇symx pTASEP,ε

t (h,≤g)| is given by the probabilitythat ht(y;−g) ≤ −h(y) for all y ∈ εZ, but for some y, ht(y;−g) > −hx(y). Clearly again y = x

and ht(x;−g) = −hx(x) + ε1/2 or +2ε1/2. Since −h(x) = −hx(x) + 2ε1/2, again by the parityconservation of the model, the last statement is the probability that ht(y;−g) + h(y) ≤ 0 for all y andht(x;−g) + h(x) = 0; or ht(y;−g) + h(y) ≤ −ε1/2 for all y and ht(x;−g) + h(x) = −ε1/2. Thatis, |∇sym

x pTASEP,εt (h,≤g)| is given by PTASEP(arg max{ht(y;−g) + h(y)} = x, ht(x;−g) + h(x) ∈

{−ε1/2, 0}). As in the last paragraph, summed over all h(0) ∈ ε1/2Z for each fixed h, we getPTASEP(arg max{ht(y;−g) + h(y)} = x).

Considering the possibility of multiple points of maxima, we have (B.1) when we sum over allx ∈ εZ. �

Thus the square of the left hand side of (B.1) is bounded by ETASEP[(X(ht, h))2]. Till now h wasfixed; now we take an expectation over µε(dh), so that the left-hand side of (5.3) is bounded byE[(X(ht, h))2] =

∑k2P(X(ht, h) = k), where P = PTASEP × µε denotes the product measure of the

TASEP dynamics and the random walk measure µε for h, and E denotes the corresponding expectation.Conditioning on ht(·;−g), to get (5.3) it is enough to show

Lemma B.2. For all ε > 0 and k ≥ 1 and any fixed function f(·) ∈ Sε such that f(x) + h(x)→ −∞as |x| → ∞, there are absolute constants C, c > 0 (not depending on ε, f, k) such that

µε(X(f, h) = k) ≤ Ce−ck1/4 . (B.2)

Note that as −g(x) ≤ −C(1 + |x|), one has that for any fixed ε > 0 almost surely the TASEPheight function at time t, ht(x;−g) + h(x)→ −∞ as |x| → ∞. In fact, one can get the same linearbounds with a slightly worse constant and a height shift. To see this, note that we can bound the initialdata above by both −C(1 + x) or −C(1 − x). By adjusting the drift, we can put a random walkheight function above each of these. Because the TASEP height function preserves order, at a latertime the height function will be dominated by the evolution of either one of these random walk initialdata. Besides the height shift, these two are equilibria, and so we conclude that our height function isagain bounded by the two random walks with drift at time t. These, in turn can be bounded above byC ′(1− x) and C ′(1 + x).

The proof of Lem. B.2 is adapted from [15] who consider f ≡ 0 on a finite interval. Intuitively,f ≡ 0 is the worst case, and it is plausible one could avoid what follows by making this intuitionrigorous.

Proof of Lem. B.2. Fix k > 1 and any function f ∈ Sε such that f(x) + h(x) → −∞ as |x| → ∞.Because of the decay condition on f + h, all maximums are attained in a compact interval almost surely,that is, there exists Lk > 0 such that all points of maxima for f + h are in [−Lk, Lk] with probability atleast 1− e−k. Let us call this event Ak. Henceforth, we restrict ourselves to points only in the compactset [−Lk, Lk]. First observe that h+ f can attain its maxima at consecutive points. Let Ek denote theevent that there is a string of

√k consecutive points at which f + h attains the maxima. Then we show

that µε(Ek) ≤ Ce−c√k for some absolute constants C, c > 0.

To this end, on the event Ek, let {x∗, x∗ + ε, . . . , x∗ + bε} be the last string of length at least√k

at which f + h attains the maxima. We consider the last√k/3 points of this string, that is the points

{x∗+ ε(b−√k/3 + 1), . . . , x∗+ εb} (to avoid cumbersome notations, we assume

√k/3 is an integer).

With u as defined in (2.4) andm the maximum value of f+ h, we have thatm := h(x)+f(x) = h(x+

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 19

ε)+f(x+ε), that is u(x) = f(x)−f(x+ε) for all x ∈ I := {x∗+ε(b−√k/3), . . . , x∗+ε(b−2)}.

That is, given the function f , the values of u(x) for all x ∈ I are fixed. Now for each such configurationof u in E , we construct 2

√k/3−2 new and distinct configurations by assigning different values of

u(x) for all x ∈ I and setting u(x∗ + ε(b −√k/3), u(x∗ + ε(b − 1)) such that for each of these

new configurations the values of h+ f at x∗ + ε(b−√k/3), x∗ + bε are different from those of the

original configuration. If we define this map as φ, then for each configuration u ∈ Ek, φ(u) is a setof 2

√k/3−2 elements and the sets are disjoint for different configurations u. To see the disjointness,

observe that it is easy to recognize the map φ−1. For any u∗ ∈ φ(u) and the walk h∗ defined fromthe sequence u∗ as h∗(x + ε) = h∗(x) + u∗(x), either (h∗ + f)(x) > m for some x − ε ∈ I or forx = x∗ + bε, in which case the number of points of maxima of h∗ + f is at most

√k/3, and we can

recover u by identifying the last string of length at least 2√k/3 of consecutive points before a point of

maxima where h∗ + f is flat. On the other hand, if (h∗ + f)(x) ≤ (h+ f)(x) for all x− ε ∈ I and(h∗ + f)(x∗ + bε) < (h+ f)(x∗ + bε), the number of points of maxima of h∗ + f is at at least 2

√k/3

and we can recover u by identifying the last string of points of maxima of length at least 2√k/3. This

gives thatµε(Ek ∩ Ak) ≤ 2−

√k/3+2 .

Now on the event Eck∩{X(f, h) = k}, there are at least√k many disjoint blocks of point of maxima

for h+ f , each of size less than√k. That is, if {xi, xi + ε, . . . , xi + (bi − 1)ε} for i = 1, 2, . . . , ` are

the points of maxima for h+ f , where b1 + b2 + . . .+ b` = k and xi + biε < xi+1, bi <√k for all i

and hence ` ≥√k. First assume that, in addition to the above, there are at least k1/3 blocks such that

in each block, there is an x such that u(x− ε) = −ε1/2 (clearly then f(x− ε) = ε1/2). Let us call thisevent E1k and enumerate these blocks as before, with ` ≥ k1/3 this time. Now for any i = 1, 2, . . . , k1/4,and any configuration u ∈ E1k , we define a new configuration Φi(u) = u∗ as follows. Let yi be the firstx such that x+ ε is in the (`− i)-th block and u(yi) = −ε1/2. Define u∗(yi) = ε1/2 and u∗(x) = u(x)

for all other x; and define the walk h∗ from the sequence u∗ as h∗(x + ε) = h∗(x) + u∗(x). Then(h∗ + f)(x) = m+ 2ε1/2 for x ∈ {xj , xj + ε, . . . , xj + (bj − 1)ε} for all `− i+ 1 ≤ j ≤ ` and forall points in the (`− i)-th block after yi. Let Φj1,j2,...,jm := Φj1 ◦ . . . ◦ Φjm and denote the set

Φ(u) = {ΦS(u) : S ⊆ {1, 2, . . . , k1/4}}

for each u ∈ E1k . Thus Φ(u) has 2k1/4

elements and for different configurations u the sets Φ(u) aredisjoint. Observe that the disjointness follows because we can identify u from u∗ by first finding themaximum value of m∗ attained for the first time at x∗ say, and then the highest value less than m∗ thatis attained in at least k1/3/2 many blocks of points less than x∗. Hence µε(E1k ) ≤ 2−k

1/4. Similarly if

there are at least k1/3-many blocks such that in each block, there is an x such that u(x) = ε1/2 (clearlythen f(x) = −ε1/2 then), and we call this event E2k , then reversing the sequence u and switching theε1/2 to −ε1/2 and vice versa, we see as before that µε(E2k ) ≤ 2−k

1/4. On the event that there does not

exist such k1/3 blocks with x such that u(x) = ε1/2 or k1/3 blocks with x such that u(x) = −ε1/3,it means that there are at most 2k1/3 blocks of size at least 2, that is, there are at least k/2 singletonpeaks, that is blocks of size 1 (since k − 2k1/3 ≥ k/2 for k large). We call this event E3k . Note thaton this event, there exists a sequence of at least

√k consecutive blocks of maxima, all of size 1. If

for a sequence u of finite length, there are√k points of maxima and each maxima is attained as a

single peak (that is block of size 1), then by defining the map ψi as ψi(u) = u∗ where u∗(xi) = ε1/2

and u∗(x) = u(x) for all other x, where xi is the i-th point of maximum counted from the rightand ψ(u) = {ψS(u) : S ⊆ {1, 2, . . . , k1/3}}, we see that ψ(u) has 2k

1/3elements and for different

configurations u the sets ψ(u) are disjoint. This implies that µε(E3k ) ≤ 2−k1/3

. Thus

µε({X(f, h) = k} ∩ Eck ∩ Ak) ≤ 2−k1/4+2.

Putting all this together we have

µε(X(f, h) = k) ≤ µε({X(f, h) = k} ∩ Eck ∩ Ak) + µε(Ek ∩ Ak) + µε(Ack) ≤ Ce−ck1/4

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 20

for some absolute constants C, c > 0. �

APPENDIX C. PROOF OF LEM. 4.2

We recall the proof in [32] as recounted in [29], and describe the minor modifications necessary toinclude the height h(0). We start with a definition. An irreducible cycle C is a sequence of integersy0, . . . , yk with y0 = yk = 0 and yi 6= yj for any other i 6= j. πC(aj) = 1

k for aj = yj − yj−1 andzero otherwise. Then it is shown in [32] that any p(·) with mean 0 and finite support can be written as afinite sum p(x) =

∑iwiπCi(x) where wi > 0,

∑iwi = 1 and Ci are irreducible cycles. From this

one obtains the representation

Mf =∑i

∑x∈Z

MCi+x MC+x = 1k

k−1∑i=0

η(x+ yi)(1− η(x+ yi+1))∇x+yi,ai+1f (C.1)

We have a Markov generator A = MC+x acting on functions on particle configurations η ∈ {0, 1}C onC := {y0+x, . . . , yk−1+x}. We can furthermore restrict to the subset U` of particle configurations witha fixed number ` of particles and our measure just becomes the uniform measure. Let A = 1

2(A+A∗)denote the symmetrization of A. It has the same range as A, namely mean zero functions (since theuniform measure is uniquely invariant for both). Since the configuration space is finite, there is a finiteB such that ∑

η∈U`

Ag(η)(−A)−1Ag(η) ≤ B2∑η∈U`

g(η)(−Ag)(η). (C.2)

Equivalently, by Cauchy-Schwarz inequality, for any α > 0,∑η∈U`

f(η)Ag(η) ≤ Bα2

∑η∈U`

f(η)Af(η) + B2α

∑η∈U`

g(η)Ag(η). (C.3)

Now let f and g be functions of h(0) and η ∈ {0, 1}Z. We can think of general function f and gof h(0) ∈ Z and η ∈ {0, 1}Z of being first of all a function of η |U` , then of `, then of all the othervariables h(0) and η(x), x ∈ Z \ C . The inequality (C.3) clearly holds, with these general functions,and the dependence on the extra variables ` and η(x), x ∈ Z \C there but not written. Nothing in (C.3)affects the variables η(x), x ∈ Z \ C . In terms of h(0), there are two cases. If all elements of C areeither in {0, 1, . . .} or in {. . . ,−2,−1} then no move of A affects h(0). Averaging the inequality overthe measure ν gives in this case∫

fMC+xgdν ≤ Bα2

∫f(−MC+xf)dν + B

2α

∫g(−MC+xg)dν. (C.4)

In the second case we have elements of C in both {0, 1, . . .} and in {. . . ,−2,−1}, so some movesaffect h(0). We claim the same inequality holds. For suppose yi + x ≤ −1 and yi+1 + x ≥ 0. Thecorresponding term in each term of our inequality on U` reads either∑

η∈U`

f(h(0), η)ηyi+x(1− ηyi+1+x)(g(h(0)− 2, ηyi+x,yi+1+x)− g(h(0), η)) (C.5)

on the left hand side, or the same thing with two f ’s or two g’s on the right hand side. We can take thesum over h(0) ∈ Z and pass it through the finite sum over U`, then take the expectation with respect tothe marginal distribution of ` and the η(x), x ∈ Z \ C under ν. It is clearly the same if we had insteadyi + x ≥ 0 and yi+1 + x ≤ −1, with the −2 replaced by a +2. Doing this to all the terms shows that(C.4) holds in the second case as well.

Now we can sum (C.4) over x ∈ Z and Ci with weights wi to obtain∫fMgdν ≤ Bα

2

∫f(−Mf)dν + B

2α

∫g(−Mg)dν. (C.6)

Optimizing over α > 0 gives (3.5).

CONVERGENCE OF EXCLUSION PROCESSES AND KPZ EQUATION TO THE KPZ FIXED POINT 21

Acknowledgements. JQ would like to thank Bálint Virág for enlightening discussions. Both au-thors were supported by the Natural Sciences and Engineering Research Council of Canada.

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(J. Quastel) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, 40 ST. GEORGE STREET, TORONTO,ONTARIO, CANADA M5S 2E4

Email address: quastel@math.toronto.edu

(S. Sarkar) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, 40 ST. GEORGE STREET, TORONTO,ONTARIO, CANADA M5S 2E4

Email address: sourav.sarkar@utoronto.ca