Long and short time laws of Iterated Logarithms for the KPZ fixed point

The Kardar-Parisi-Zhang (KPZ) universality class consists of a broad family of random growth models that are believed to exhibit certain common features, such as universal scaling exponents and limiting distributions. Several models such as asymmetric simple exclusion processes, last passage percolation, directed polymers in random environment, driven lattice gas models, KPZ equation are believed to lie in this class. All these models are conjectured to have an universal limiting structure, namely the KPZ fixed point under the so called ‘KPZ’ scaling.

Over the last four decades, an immense progress has been achieved in understanding a large collection of models in the KPZ universality class (see [Qua11, Cor12, QS15, HHNT15, Tak18]) by intermingling the ideas from stochastic PDEs, random matrix theory, representation theory of the quantum groups and various other research directions. The universal scaling limit, the KPZ fixed point, itself was rigorously first constructed in [MQR21] as a scaling limit of TASEP by specifying the transition probabilities. An alternative description of the KPZ fixed point as a variational formula involving space-time Airy sheet was conjectured in [CQR15]. This space-time Airy sheet, later known as the directed landscape (modulo a parabolic shift), was derived rigorously as a scaling limit of Brownian last-passage percolation in [DOV18]. The connection between the two objects was obtained in [NQR20], where the authors proved the conjectural variational formulation of the KPZ fixed point, providing a coupling of all initial data on the same probability space based on the directed landscape.

In this paper, we introduce the KPZ fixed point using the variational formulation which involves the directed landscape. We introduce the latter below.

Initial data for the KPZ fixed point plays an important role in the precise nature of the fluctuations of the height function. Owing to exact solvability in some prelimiting KPZ models, the KPZ fixed point marginals for the three fundamental initial data: narrow wedge, flat, and Brownian, has been well studied in the literature.

In this paper we consider the KPZ fixed point started from a suitable class of initial data which includes the above three fundamental initial data.

We investigate the evolution of the heights of the large peaks of the KPZ fixed point in global and local scale through the lens of law of iterated logarithms (LIL). More precisely, we study LIL for the height function of the KPZ fixed point $\mathfrak{h}_{t}(0)$ at spatial location $x=0$ as $t\uparrow\infty$ (long-time LIL), as well as the temporal increment of the KPZ fixed point near $1$, i.e., $\mathfrak{h}_{t}(0)-\mathfrak{h}_{1}(0)$ as $t\downarrow 1$ (short-time LIL). We first state our long-time LIL result below.

The above result identifies the limiting height of the peaks of the KPZ fixed point. Note that constant in (1.4) is same for all initial data satisfying (1.3). This naturally leads to the following question:

As in other cases like Brownian motion or the Kardar-Parisi-Zhang equation [DG21], the value of the constant in the law of iterated logarithms of the KPZ fixed point is governed by the tail behavior of the one point distribution. As it turns out, for the initial data considered above, the (upper)-tail behavior of $t^{-1/3}\mathfrak{h}_{t}(0)$ is similar to the (upper)-tail behavior of Tracy-Widom GUE distribution. If $X$ is a Tracy-Widom GUE distributed random variable, the upper tail, $\mathbb{P}(X\geq s)$, roughly behaves like $e^{-4s^{3/2}/3}$ for large $s$. The $3/2$ exponent in the tail induces a $(\log\log t)^{2/3}$ type behavior in the law of iterated logarithm whereas the constant $(3/4)^{2/3}$ comes from the leading coefficient in the exponent.

As depicted in Theorem 1.3, the initial data can atmost grow as $\sqrt{x}$. We now explain the reason behind such a choice of initial data. It is well known that the directed landscape $\mathcal{L}(z,0;0,t)$ intuitively behaves like $-z^{2}/t$ (see Section 2.2). Thus following (1.2) we heuristically expect $\mathfrak{h}_{t}(0)$ to grow like as

Note that $\mathfrak{h}_{t}(0)$ is of the order $t^{1/3}$ as long as $\mathfrak{h}_{0}(x)\leq\mathrm{C}_{\operatorname{sq}}\sqrt{1+|x|}$. For initial data with higher growth, one needs to subtract a non-trivial growing mean term from $\mathfrak{h}_{t}(0)$ to see $t^{1/3}$ order fluctuations. Up to such centering, we believe that similar result as in (1.4) would hold. However, we defer this case to some future work. Theorem 1.3 leaves out the Brownian initial data. We now state the corresponding result for Brownian data separately.

Note that although $(\log\log t)^{2/3}$ scaling above is same as Theorem 1.3, the limiting constant in (1.5) is different compared to (1.4). This is due to the fact the distribution of $t^{-1/3}\mathfrak{h}_{t}(0)$ for Brownian data is given by the Baik-Rains distribution which has $e^{-2s^{3/2}/3}$ upper tail decay [FV21] in comparison to the tail decay $e^{-4s^{3/2}/3}$ for the Tracy-Widom GUE distribution. This leads to the above $(3/2)^{2/3}$ constant.

Theorems 1.3 and 1.4 provide precise long-time LIL for the KPZ fixed point for a large class of data. However, it leaves out the case of liminf type law of iterated logarithm which corresponds to the limiting depth of the valleys of the KPZ fixed point. Such liminf result is related to how the lower tail probability $\mathbb{P}(t^{-1/3}\mathfrak{h}_{t}(0)\leq-s)$ decays with $s$. For a large class of initial data, the lower tail probability decays like $e^{-\gamma s^{3}}$ for large $s$. Here $\gamma:=\frac{1}{12}$ for narrow wedge and Brownian data and $\gamma:=\frac{1}{6}$ for flat initial data (see Example 1.16 in [QR19]). Thus we expect

However, it is not known what would be the constant $\gamma$ in the lower tail for all initial data satisfying 1.3. As a result, it is less clear what should be the limiting constant for the liminf case. Once the value of the constant is determined, the proof of (1.6) will be very similar to that of Theorem 1.3.

We next turn towards our short-time LIL result that identifies local temporal growth behavior.

We see that local growth of the KPZ fixed point in the temporal direction is of the order $\varepsilon^{1/3}(\log\log\varepsilon^{-1})^{2/3}$. Furthermore, the precise constant $(3/2)^{2/3}$ is universal in the sense that it does not depend on the initial data. We now briefly explain why this is the case. Indeed note that due to the metric composition law in (1.1) and the variational formula in (1.2) we have

By the scale invariance properties of the directed landscape, $\varepsilon^{-\frac{1}{3}}\mathcal{L}(\varepsilon^{\frac{2}{3}}x,1;0,1+\varepsilon)$ is equal in distribution to $\mathcal{A}(x)-x^{2}$ where $\mathcal{A}(\cdot)$ is the stationary $\operatorname{Airy}_{2}$ process. On the other hand, by local Brownianity of the KPZ fixed point (see Theorem 4.14 in [MQR21]), for any initial data of Theorem 1.5, we have

where $\mathfrak{B}(x)$ is a two sided Brownian motion with diffusion coefficient $2$. Thus, as $\varepsilon\downarrow 0$, we expect $\nabla\mathfrak{h}_{\varepsilon}$ converge to the Baik-Rains distribution [BR00] which is defined as $\sup_{x\in\mathbb{R}}(\mathfrak{B}(x)+\mathcal{A}(x)-x^{2})$. The constant $(3/2)^{2/3}$ appearing in (1.8) can then be anticipated from the sharp upper tail asymptotics of the Baik-Rains distribution [FV21].

Note that Theorem 1.5 includes a larger class of functional data compared to Theorem 1.3. This is partly because we are interested in the local growth behavior of the KPZ fixed point around $t=1$. As we explain below, studying local growth of the KPZ fixed point does not require a non-trivial centering of $\mathfrak{h}_{1}(0)$ for a larger class of initial data as opposed to the case in Theorem 1.3. To see this, note that the directed landscape $\mathcal{L}(z,0;0,1)$ decays like $-z^{2}$. Thus following (1.2) we expect $\mathfrak{h}_{1}(0)$ to be centered around

The last expression is finite whenever $\mathfrak{h}_{0}(x)\leq\mathrm{A}+(1-\kappa)x^{2}$. This justifies the condition in (1.7).

Like as the limsup result of Theorem 1.5, the height increments of the KPZ fixed point near $t=1$ also admit a liminf LIL as the time increments converges to $0$. More precisely, for the KPZ fixed point started from any initial data considered in Theorem 1.5, we expect

The proof of the above fact can be completed using similar ideas as in the proof of Theorem 1.5. We defer showing this result to a future work.

Theorem 1.5 studies the local growth of the KPZ fixed point along temporal direction near the time $1$. However the same result holds for any other fixed time $t$ (assuming $\mathfrak{h}_{0}(x)\leq\mathrm{A}+(1-\kappa){x^{2}}/{t}$ instead of (1.7) for functional data) and the proof follows in the same way. It is worthwhile to ask if the same result hold for all $t\in[c,d]$ uniformly. In other words, how often does the short time LIL for the KPZ fixed point fails to hold on an interval. Similar questions were studied for the Brownian motion in [OT74, Theorem 2] where the authors had showed the Hausdorff dimension of the set of time points where the LIL fails is almost surely equal to $1$. In fact, their result is much deeper. Taking inspiration from their result and drawing on the analogy between the short time LILs of the KPZ fixed point and the Brownian motion, we make the following conjecture.

.

We now briefly describe the proof idea of our main results. We first explain the details for the short-time LIL case. The spirit of the arguments for the long time LIL is similar and will be touched upon later in this section.

Let us recall the notation $\nabla\mathfrak{h}_{\varepsilon}$ from (1.9). As alluded in the discussion after (1.9), as $\varepsilon\downarrow 0$, we expect $\nabla\mathfrak{h}_{\varepsilon}$ converges to the Baik-Rains distribution. Although we have precise tail estimates of the Baik-Rains distribution, to conclude the law of iterated logarithm type result, we need to produce sharp uniform tail estimates for $\nabla\mathfrak{h}_{\varepsilon}$. One of the main contributions of this paper is to establish such estimates.

Let us briefly explain the proof idea behind the above result. Towards this end, let us define

where $\mathcal{A}$ is independent of $\mathfrak{h}_{1}$. The proof of the above proposition proceeds in two steps:

1. (i)

   There exists an absolute constant $b>0$ such that $\mathbb{P}(\nabla\mathfrak{h}_{\varepsilon}\neq\nabla\mathfrak{h}_{\varepsilon}^{[-b\sqrt{s},b\sqrt{s}]})\leq\exp(-\frac{2}{3}s^{3/2})$.

2. (ii)

   Proving Proposition 1.7 with $\nabla\mathfrak{h}_{\varepsilon}$ replaced by $\nabla\mathfrak{h}_{\varepsilon}^{[-b\sqrt{s},b\sqrt{s}]}$.

We first describe how to show item (ii). For instructive purposes, here we explain how to derive it for $\nabla\mathfrak{h}_{\varepsilon}^{[0,b\sqrt{s}]}$. Set $a:=\varepsilon^{2/3}b\sqrt{s}$. Let us denote

where $\mathfrak{B}$ is an independent two sided Brownian motion with diffusion coefficient $2$. Let us assume for simplicity the above argmax are uniquely defined almost surely. The key idea behind our proof is a Brownian replacement principle that will allow us to compare the KPZ fixed point starting from a general initial data with that starting from the Brownian initial data on certain events. Indeed, owing to the geometric properties of the KPZ fixed point, [Pim21] argues for any $\mu>0$ on the event $\mathsf{E}^{\mu}(a):=\{Z(a)\leq Z^{\mu}(-a),Z(-a)\geq Z^{-\mu}(a)\}$ one has

where we set $\mathfrak{h}_{1}^{\mu}(x):=\sup_{z\in\mathbb{R}}(\mathfrak{B}(z)+\mu x+\mathcal{L}(z,0;x,1))$ for all $x\in\mathbb{R}$.

We call the above trick by the name Brownian replacement trick which was first discovered in the present form by Leandro Pimentel [Pim14]. It is essentially a property of the basic coupling in last passage percolation models. On $\mathsf{E}^{\mu}(a)$, we can then transfer our analysis to the KPZ fixed point started with Brownian motion with drift as initial data. The advantage of working with this initial data is that such data is known to be stationary for the KPZ fixed point (see Theorem 4.5 [MQR21]). In particular we have $\mathfrak{h}_{1}^{\mu}(x)-\mathfrak{h}_{1}^{\mu}(0)\stackrel{{\scriptstyle d}}{{=}}\mathfrak{B}(x)+\mu x$. Under the diffusive scaling appearing in (1.9), $\varepsilon^{-1/3}(\mathfrak{h}_{1}^{\mu}(\varepsilon^{2/3}x)-\mathfrak{h}_{1}^{\mu}(0))$ is then equal in distribution to $\mathfrak{B}(x)+\varepsilon^{1/3}\mu x$. As $x$ varies in $[0,b\sqrt{s}]$, the drift term is bounded by $\varepsilon^{1/3}\mu b\sqrt{s}$. Thus on $\mathsf{E}^{\mu}(a)$,

Taking $\mu=\varepsilon^{-1/4}\sqrt{s}$, one can ensure the drift term above is negligible compared to $s$. In fact, for this choice of $\mu$ and $a$, we have $\mathbb{P}(\mathsf{E}^{\mu}(a)^{c})\leq\exp(-\frac{2}{3}s^{3/2})$. To prove this, one relies on exponential type tail estimates for the argmax $Z(\pm a),Z^{\mu}(\pm a)$ that are developed in Section 3 using parabolic decay estimates of directed landscape (Lemma 2.5).

Thus (1.10) for $\nabla\mathfrak{h}_{\varepsilon}^{[0,b\sqrt{s}]}$ now follows from upper tail asymptotics of Baik-Rains distribution which is given by [FV21]. An analogous argument provides the same upper tail estimate for $\nabla\mathfrak{h}_{\varepsilon}^{[-b\sqrt{s},0]}$. This justifies item (ii). Item (i) appears as Lemma 3.5 later. Its proof also involves the above Brownian replacement trick shown and tail estimates for the argmax. For brevity, we skip the details here and encourage the readers to read the details in Section 3.

The above description of the main argument for proving (1.10) is of course quite reductive, and the full argument, presented in Section 4.3, relies on various technical estimates related to the location of the argmax that are discussed in Section 3. Apart from the temporal increment tail, these estimates on the argmax also allow us to extract other probabilistic information or the KPZ fixed point such as growth estimates (Proposition 4.1) and spatial modulus of continuity (Proposition 4.4).

In fact, the precise statement of Proposition 4.5 can handle absolute value of any small scaled increment in the vicinity of $1$. As a consequence, this leads to suitable temporal modulus of continuity type estimates in Corollary 4.6 and Proposition 4.7. Combining the temporal modulus of continuity estimates along with sharp upper tail asymptotics of the increments leads to the upper bound for the short-time LIL.

The lower bound for the short-time LIL on the other hand relies on showing a certain kind of independence structure in the temporal increments of the KPZ fixed point. Loosely speaking we show that for two small increments $\varepsilon_{1}\ll\varepsilon_{2}\ll 1$, $\nabla\mathfrak{h}_{\varepsilon_{1}}$ and $\nabla\mathfrak{h}_{\varepsilon_{2}}$ are approximately independent. Keeping $\nabla\mathfrak{h}_{\varepsilon_{2}}$ as it is, we construct a proxy for $\nabla\mathfrak{h}_{\varepsilon_{1}}$ suitable for comparison with $\nabla\mathfrak{h}_{\varepsilon_{2}}$. Towards this end we define

In plain words, we obtain $\nabla\mathfrak{h}_{\varepsilon_{1}\downarrow\varepsilon_{2}}$ from $\nabla\mathfrak{h}_{\varepsilon_{1}}$ by naïvely replacing the directed landscape $\mathcal{L}(\varepsilon_{1}^{2/3}x,1;0,1+\varepsilon_{1})$ by $\mathcal{L}(\varepsilon_{1}^{2/3}x,1+\varepsilon_{2};0,1+\varepsilon_{1})$ (and hence the name ‘Landscape Replacement’). Note that the proxy $\nabla\mathfrak{h}_{\varepsilon_{1}\downarrow\varepsilon_{2}}$ is same as $\nabla\mathfrak{h}_{\varepsilon_{1}-\varepsilon_{2}}$ in distribution and furthermore the landscape part of the proxy is independent from that of $\nabla\mathfrak{h}_{\varepsilon_{2}}$. As $\frac{\varepsilon_{1}}{\varepsilon_{2}}\to\infty$, this proxy turns out to be very good approximation of $\nabla\mathfrak{h}_{\varepsilon_{1}}$ in the following sense.

Due to the presence of $\frac{\varepsilon_{1}}{\varepsilon_{2}}$ term on the r.h.s. of (1.12), this suggests $\nabla\mathfrak{h}_{\varepsilon_{1}}$ and $\nabla\mathfrak{h}_{\varepsilon_{2}}$ are approximately independent when $\varepsilon_{1}$ and $\varepsilon_{2}$ are far apart on a multiplicative scale, i.e., $\frac{\varepsilon_{1}}{\varepsilon_{2}}\gg 1$. This suggests the decorrelation of the temporal increments happens at a slow rate and thus produces a law of iterated logarithm behavior.

Notice that the above theorem only ensures the landscape part of $\nabla\mathfrak{h}_{\varepsilon_{1}}$ and $\nabla\mathfrak{h}_{\varepsilon_{2}}$ appearing in the variational problem in (1.9) are approximately independent. We now briefly explain how the rescaled spatial processes appearing in corresponding variational problems are also approximately independent. This part of the argument proceeds similarly to the proof of Proposition 1.7 explained above. Indeed, thanks to the Brownian replacement trick explained earlier, we can again transfer to Brownian motion with drifted initial data. The drift term can again be ignored due to the presence of diffusive scaling. Restricting the supremum to appropriate disjoint intervals and using independence of the increments for Brownian motion, one can obtain an independence structure within the temporal increments of the KPZ fixed point. The details of the proof of short time landscape replacement and the extraction of lower bound from it are presented in Section 5.2 and Section 6.2.2 respectively.

For the long-time LIL, one of the key input is the sharp one point tail estimates for $t^{-1/3}\mathfrak{h}_{t}(0)$, available from [FV21]. Combining this with the temporal modulus of continuity estimates (Proposition 4.5) discussed earlier, produces the upper bound for the long-time LIL result. For the lower bound, we rely on a similar landscape replacement trick as discussed above. Given two temporal KPZ fixed point height functions, $\mathfrak{h}_{t_{1}}(0)$ and $\mathfrak{h}_{t_{2}}(0)$ with $t_{2}\gg t_{1}$, as before we construct a proxy for $\mathfrak{h}_{t_{2}}(0)$ by suitably changing the directed landscape part of the variational problem in (1.2). More precisely, we define the proxy to be

Just as in Theorem 1.8, this proxy turns out to be good approximation $\mathfrak{h}_{t_{2}}(0)$ in the following sense.

Proof of the above result relies on precise estimate on the location of the argmax for the variational problem in (1.2). This estimate is captured in Proposition 3.2. In the case of non-random initial data (i.e., $\mathfrak{h}_{0}$ is non-random), the proxy $\mathfrak{h}^{t_{2}\downarrow t_{1}}$ is independent of $\mathfrak{h}_{t_{1}}(0)$. For Brownian initial data, the independence structure is not apriori clear since $\mathfrak{h}^{t_{2}\downarrow t_{1}}$ and $\mathfrak{h}_{t_{1}}(0)$ are coupled via the initial data $\mathfrak{h}_{0}(\cdot)$. However, an almost independence structure can be obtained by restricting the supremum on appropriate disjoint intervals. We refer the readers to Section 5.1 and Section 6.1.2 for the technical details of the above conceptual argument.

As a passing comment, we mention that this landscape replacement idea previously appeared in a different guise for the KPZ equation with narrow wedge initial data in [CGH21, DG21]. For instance, [DG21] (see Proposition 6.1) developed an analogue of landscape replacement trick for showing the law of iterated logarithms for the KPZ equation as time variable goes to $\infty$. Two main tools in [DG21] are $(1)$ the convolution formula for the KPZ equation which is often regarded as the positive temperature analogue of the variational description of the KPZ fixed point and, $(2)$ the $\mathbf{H}$-Brownian Gibbs property of KPZ line ensemble [CH16]. While the convolution formula of the KPZ equation is valid starting from any initial data, the $\mathbf{H}$-Brownian Gibbs property fails to hold when KPZ equation is started from an initial data which is not narrow wedge. This restricts [DG21] to extend their results beyond the narrow wedge initial condition. In contrast to [DG21], the long time LILs of Theorem 1.3 and 1.4 covers a large class of initial data by applying landscape replacement trick followed by using rich geometric structure of the directed landscape. On the other hand, the short time LIL of Theorem 1.5 relies on the aforementioned Brownian replacement trick instead. This latter trick which reflects locally Brownian nature of the KPZ fixed point is very much different than the replacement tricks used in [DG21]. In fact, the Brownian replacement trick allows us to handle increment of the spatial profile for a large class of initial data just based on one point tail estimates of the location of argmax in the variational formula of (1.2), a property which is currently beyond reach for the KPZ equation or any other positive temperature models in the KPZ universality class.

Our main results on the growth of tall temporal peaks and local temporal growth of the KPZ fixed point draw inspirations from a series of efforts in understanding the geometry of the KPZ fixed point and the directed landscape. Being an universal scaling limit, the KPZ fixed point enjoys several rich properties and exhibits remarkable fractal geometry. For example, for all sub-parabolic initial data, $\mathfrak{h}_{t}(x)$ is locally absolutely continuous with respect to Brownian motion for each $t>0$, and Hölder $1/3^{-}$ continuous in $t$ [CH14, SV21]. Thus for rapidly decaying initial data, for each $t>0$ the $\mathfrak{h}_{t}(\cdot)$ has a unique maximizer almost surely [MFQR13, CH14, Pim14, SV21, CHHM21]. In this context, a recent line of inquiries includes the investigation of the fractal nature of exceptional times where $\mathfrak{h}_{t}(\cdot)$ has multiple maximizers [CHHM21, Dau22].

Fractal geometry of the directed landscape and its geodesics has also been studied extensively in recent years (see [Gan21] for a partial survey). In [BGH21] the author initiated the study of the difference profile of the directed landscape $\mathcal{D}(x,t):=\mathcal{L}(1,0,x,t)-\mathcal{L}(-1,0,x,t)$. Difference profiles exhibit rich fractal behavior [BGH21, GZ22], has connection to disjointness of directed geodesics [BGH19], and are related to Brownian local time [GH21]. Certain qualitative features of geodesics are also explored [DSV20].

Apart from the KPZ fixed point, long time law of iterated logarithms has also been rigorously shown in certain pre-limiting models in the KPZ universality class. For exponential last passage percolation, [Led18, BGHK19] identified correct scaling of the law of iterated logarithms for the $\limsup$ and $\liminf$ of the rescaled passage time. In the context of random matrices, [PZ17] proved a law of fractional logarithm for the $\limsup$ of the sequence of eigenvalues in a GUE minor process.

For the KPZ equation, the precise long time LIL in the temporal direction was obtained in [DG21]. Apart from the results related to LILs, [DG21] also demonstrated certain fractal behavior of the level sets of the peaks for the KPZ equation. Understanding fractal nature of such level sets is important in the context of non-linear SDEs as it is intimately connected with the phenomenon of intermittency (see [KKX17, KKX18, DG21, GY21] are references there in).

Short time LIL for the temporal increment of the KPZ equation has been recently proved in [Das22]. The exponents for such LIL for the KPZ equation is marked different from our Theorem 1.5 as the local temporal structure for the KPZ equation is governed by a fractional Brownian motion of index $\frac{1}{4}$ (see [KSXZ13, Das22]).

The rest of the paper is organized as follows. In Section 2 we collect several interesting properties and estimates for the directed landscape and the KPZ fixed point. In Section 3 we determine quantitative estimates for the location of the supremum of the variational problem of the KPZ fixed point. These estimates appear in Proposition 3.2, Corollary 3.3 and Proposition 3.5. In Section 4, we establish growth estimates (Proposition 4.1), spatial modulus of continuity (Proposition 4.4), and and temporal modulus of continuity (Corollary 4.6) estimates for the KPZ fixed point. In Section 5 we prove our long time and short time landscape replacement theorems that provide independent proxies for the KPZ fixed point (see the explanation in Section 1.3). Finally in Section 6 we prove our main results: Theorem 1.3, Theorem 1.4, and Theorem 1.5. Proof of two technical lemmas are deferred to Appendix A.

The authors thank Ivan Corwin for helpful comments on an earlier draft of the paper. The authors acknowledge support from NSF DMS-1928930 during their participation in the program “Universality and Integrability in Random Matrix Theory and Interacting Particle Systems” hosted by the Mathematical Sciences Research Institute in Berkeley, California in fall 2021. SD’s research was also partially supported by Ivan Corwin’s NSF grant DMS-1811143 and the Fernholz Foundation’s “Summer Minerva Fellows” program.

The directed landscape as defined in Definition 1.1 satisfies several symmetries. We state these results in terms of the stationary version of directed Landscape defined as

We reserve the notation $\mathcal{K}(\cdot,\cdot;\cdot,\cdot)$ for the stationary directed landscape defined in (2.1) for the rest of the article. The marginals of $\mathcal{K}$ are given by the following lemma.

Finally we discuss some useful identities for the KPZ fixed point started from narrow wedge or Brownian initial data.

In this section we prove a couple of probabilistic estimates for the directed landscape: Lemma 2.4 and Lemma 2.5. The first one collects several tail estimates for the spatial process of the directed landscape, whereas the second one establishes parabolic decay estimates for the directed landscape.