Time-time Covariance for Last Passage Percolation with Generic Initial Profile

Abstract

We consider time correlation for KPZ growth in 1 + 1 dimensions in a neighborhood of a characteristics. We prove convergence of the covariance with droplet, flat and stationary initial profile. In particular, this provides a rigorous proof of the exact formula of the covariance for the stationary case obtained in Ferrari and Spohn (2011). Furthermore, we prove the universality of the first order correction when the two observation times are close and provide a rigorous bound of the error term. This result holds also for random initial profiles which are not necessarily stationary.

Appendices

Appendix A: Bounds on Point-to-Point LPP

In the proofs, we use known results for the point-to-point LPP with exponential random variables, which we recall here.

Proposition A.1

For $\eta \in (0,\infty )$ define $\mu =(\sqrt {\eta \ell }+\sqrt {\ell })^{2}$ , $\sigma =\eta ^{-1/6}(1+\sqrt {\eta })^{4/3}$ , and the rescaled random variable

$$ L^{\text{resc}}_{\ell}:=\frac{L_{(0,0)\to(\eta\ell,\ell)}-\mu}{\sigma\ell^{1/3}}. $$

(A.1)

• (a) Limit law

  $$ \lim_{\ell\to\infty} {\mathbb{P}}(L^{\text{resc}}_{\ell}\leq s) = F_{\text{GUE}}(s), $$

  (A.2)

  with F_GUE the GUE Tracy-Widom distribution function.

• (b) Bound on upper tail: there exist constants s₀,ℓ₀,C,c such that

  $$ {\mathbb{P}}(L^{\text{resc}}_{\ell}\geq s)\leq C e^{-c s} $$

  (A.3)

  for all ℓ ≥ ℓ₀ and s ≥ s₀.

• (c) Bound on lower tail: there exist constants s₀,ℓ₀,C,c such that

  $$ {\mathbb{P}}(L^{\text{resc}}_{\ell}\leq s)\leq C e^{-c |s|^{3/2}} $$

  (A.4)

  for all ℓ ≥ ℓ₀ and s ≤−s₀.

(a) was proven in Theorem 1.6 of [30]. Using the relation with the Laguerre ensemble of random matrices (Proposition 6.1 of [1]), or to TASEP described above, the distribution is given by a Fredholm determinant. An exponential decay of its kernel leads directly to (b). See e.g. Proposition 4.2 of [25] or Lemma 1 of [3] for an explicit statement. (c) was proven in [3] (Proposition 3 together with (56)). In the present language it is reported in Proposition 4.3 of [25] as well.

Appendix B: Bounds for Point-to-Line LPP

Proposition B.1

Let ${\mathcal L}=\{(k,-k),k\in \mathbb{Z} \}$. Consider the rescaled LPP from $\mathcal L$ to (ℓ,ℓ) given by

$$ L^{\mathcal{L}, \text{resc}}_{\ell}=\frac{L_{\mathcal{L}}\to (\ell,\ell)-4\ell}{2^{4/3}\ell^{1/3}}. $$

(B.1)

• (a) Limit law

  $$ \lim_{\ell\to\infty} {\mathbb{P}}(L^{\mathcal{L}, \text{resc}}_{\ell}\leq s)=F_{\text{GOE}}(2^{2/3} s). $$

  (B.2)

• (b) Bound on upper tail: there exists constants s₀,ℓ₀,C,c such that

  $$ {\mathbb{P}}(L^{\mathcal{L}, \text{resc}}_{\ell}\geq s)\leq C e^{-c s} $$

  (B.3)

  for all ℓ ≥ ℓ₀ and s ≥ s₀.

• (c) Bound on lower tail: there exists constants s₀,ℓ₀,C,c such that

  $$ {\mathbb{P}}(L^{\mathcal{L}, \text{resc}}_{\ell}\leq s)\leq C e^{-c |s|^{3/2}} $$

  (B.4)

  for all ℓ ≥ ℓ₀ and s ≤−s₀.

(a) was obtained in [11, 48] in terms of TASEP, which can be directly rewritten in term of LPP (the complete proof is present in [10]). For general slopes of $\mathcal L$ it was shown in [27]. (b) this tails follows from the asymptotic analysis on the correlation kernel made in [10]. (c) It follows from (A.4) since ${\mathbb {P}}(L_{\mathcal {L}}\to (\ell ,\ell )\leq x)\leq {\mathbb {P}}(L_{(0,0)\to (\ell ,\ell )}\leq x)$.

Appendix C: Bounds on LPP with Random Initial Condition

Proposition C.1

Define $L^{\sigma }_{\mathcal {L}\to (\ell ,\ell )}=\max _{k} \{L_{(k,-k)\to (\ell ,\ell )}+h^{0}(k,-k)\}$ with h⁰ as in (2.6), and consider the rescaled LPP time

$$ L^{\sigma,\mathrm resc}_{\ell}=\frac{L^{\sigma}_{\mathcal{L}\to(\ell,\ell)}-4\ell}{2^{4/3}\ell^{1/3}}. $$

(C.1)

Then, there exists constants s₀,ℓ₀,C,c such that:

• (a) Bound on upper tail:

  $$ {\mathbb{P}}(L^{\sigma,\mathrm resc}_{\ell}\geq s)\leq C e^{-c s} $$

  (C.2)

  for all ℓ ≥ ℓ₀ and s ≥ s₀.

• (b) Tail on lower tail:

  $$ {\mathbb{P}}(L^{\sigma,\mathrm resc}_{\ell}\leq s)\leq C e^{-c |s|^{3/2}} $$

  (C.3)

  for all ℓ ≥ ℓ₀ and s ≤−s₀.

Proof

• (a) Define J(u) = u(2ℓ)^2/3(1,− 1) and W_ℓ(u) = h⁰(J(u))/(2^4/3ℓ^1/3). By Donsker’s theorem, u↦W_ℓ(u) converges weakly to a two-sided Brownian motion with diffusion coefficient 2σ². Further, define

  $$ L^{\text{pp}}_{\ell}(u):=\frac{L_{J(u)\to (\ell,\ell)}-4\ell}{2^{4/3}\ell^{1/3}}. $$

  (C.4)

  Then, we can write

  $$ L^{\sigma,\mathrm resc}_{\ell} = \max_{u} \{L^{\text{pp}}_{\ell}(u)+W_{\ell}(u)\} \leq \max_{u} \{L^{\text{pp}}_{\ell}(u)+u^{2}/2\}+\max_{u} \{W_{\ell}(u)-u^{2}/2\}. $$

  (C.5)

  Thus,

  $$ {\mathbb{P}}(L^{\sigma,\mathrm resc}_{\ell}\geq s)\!\leq\! {\mathbb{P}}(\max_{u} \{L^{\text{pp}}_{\ell}(u)+u^{2}/2\}\!\geq\! s/2)+{\mathbb{P}}(\max_{u} \{W_{\ell}(u)-u^{2}/2\}\geq s/2). $$

  (C.6)

  By computations based on Doob maximal inequality (used for instance in (4.36)), one obtains ${\mathbb {P}}(\max _{u} \{W_{\ell }(u)-u^{2}/2\}\geq s/2)\leq C e^{-c s^{2}}$ for some constants C,c > 0. To bound the first term without new estimates, remark that for any M we can bound

  $$\begin{array}{@{}rcl@{}} {\mathbb{P}}(\max\limits_{u} \{L^{\text{pp}}_{\ell}(u)+u^{2}/2\}\geq s/2) &\leq& {\mathbb{P}}(\max\limits_{u} L^{\text{pp}}_{\ell}(u)\geq s/4-M^{2}/2)\\ &&+ {\mathbb{P}}(\max\limits_{|u|>M} \{L^{\text{pp}}_{\ell}(u)+u^{2}/2\}\geq s/4) \end{array} $$

  (C.7)

  The exponential decay in s for the second term is just a special case of (4.19) (set τ = 0) and it holds for all M ≥ M₀, for some finite M₀. We fix M = M₀ and then, using the fact that $\max _{u} L^{\text {pp}}_{\ell }(u)=L^{\mathcal {L}, \text {resc}}_{\ell }$, by (B.3) we have exponential decay in s for the first term as well.

• (b) It follows from (A.4) since ${\mathbb {P}}(L^{\sigma }_{\mathcal {L}}\to (\ell ,\ell )\leq x)\leq {\mathbb {P}}(L_{(0,0)\to (\ell ,\ell )}\leq x)$.

□

Appendix D: Bounds on Stationary LPP

We now state and give a short proof of the tails of the one-point distribution in the stationary case with ρ = 1/2 of the LPP to (ℓ,ℓ).

Proposition D.1

Let ρ = 1/2. Then there exists constants s₀,ℓ₀,C,c such that:

• (a) Bound on upper tail:

  $$ {\mathbb{P}}({L^{\mathcal{B}}}_{(0,0)\to(\ell,\ell)}\geq 4\ell+ 2^{4/3} s \ell^{1/3})\leq C e^{-c s} $$

  (D.1)

  for all ℓ ≥ ℓ₀ and s ≥ s₀.

• (b) Bound on lower tail:

  $$ {\mathbb{P}}({L^{\mathcal{B}}}_{(0,0)\to(\ell,\ell)}\leq 4\ell+ 2^{4/3} s \ell^{1/3})\leq C e^{-c |s|^{3/2}} $$

  (D.2)

  for all ℓ ≥ ℓ₀ and s ≤−s₀.

Proof

• (a) One can write ${L^{\mathcal {B}}}_{(0,0)\to (\ell ,\ell )}=\max \{L^{\vert ,\rho }(\ell ,\ell ),L^{-,\rho }(\ell ,\ell )\}$, where L^|,ρ(ℓ,ℓ) (resp. L^−,ρ(ℓ,ℓ)) are the LPP with one-sided perturbation only on i = 0 (resp. j = 0). Then,

  $$ {\mathbb{P}}({L^{\mathcal{B}}}_{(0,0)\to(\ell,\ell)}\geq x)\leq {\mathbb{P}}(L^{\vert,\rho}(\ell,\ell)\geq x)+{\mathbb{P}}(L^{-,\rho}(\ell,\ell)\geq x). $$

  (D.3)

  By choosing x = 4ℓ + s2^4/3ℓ^1/3, Lemma 3.3 of [27] (based on the estimates on the correlation kernel in [1]) gives exponential decay in s for all s ≥ s₀.

• (b) It follows from (A.4), since ${\mathbb {P}}({L^{\mathcal {B}}}_{(0,0)\to (\ell ,\ell )}\leq x)\leq {\mathbb {P}}(L_{(0,0)\to (\ell ,\ell )}\leq x)$.

□

Lemma D.2

Let ρ = 1/2 and define I(u) = (ℓ − 2uℓ^2/3,ℓ + 2uℓ^2/3). Then, for any α > 0, we have

$$ {\mathbb{P}}(|{L^{\mathcal{B}}}_{(0,0)\to I(K)} -{L^{\mathcal{B}}}_{(0,0)\to I(0)}|\geq \alpha \ell^{1/3})\leq 4 e^{-\alpha^{2}/(16 K)} $$

(D.4)

for all ℓ large enough. Furthermore,

$$ {\mathbb{P}}(\max_{u\geq K}{L^{\mathcal{B}}}_{(0,0)\to I(u)} -{L^{\mathcal{B}}}_{(0,0)\to I(K)}-\beta u^{2} \ell^{1/3}\geq \alpha \ell^{1/3})\leq C e^{-\frac{(\alpha+\beta K^{2})^{2}}{16 K}}, $$

(D.5)

for a constant C and for all β > 0 and α > −βK² and ℓ large enough.

Proof

The process $K\mapsto Y_{K}:={L^{\mathcal {B}}}_{(0,0)\to I(K)} -{L^{\mathcal {B}}}_{(0,0)\to I(0)}$ is a martingale [7] given by a sum of i.i.d.zero mean random variables Z_j − 2, with $Z_{j}\sim \text {Exp}(1/2)$. By the exponential Chebyshev inequality,

$$\begin{array}{@{}rcl@{}} {\mathbb{P}}(|Y_{K}|\geq \alpha\ell^{1/3})& \leq& {\mathbb{P}}(Y_{K}\geq \alpha \ell^{1/3}) + {\mathbb{P}}(-Y_{K}\geq \alpha\ell^{1/3})\\ &\leq& \inf\limits_{t\geq 0} e^{-t\alpha \ell^{1/3}} {\mathbb{E}}(e^{t (Z_{1}-2)})^{2K\ell^{2/3}} + \inf\limits_{t^{\prime}\geq 0} e^{-t^{\prime}\alpha\ell^{1/3}} {\mathbb{E}}(e^{-t^{\prime}(Z_{1}-2)})^{2K\ell^{2/3}}. \end{array} $$

(C.6)

Using ${\mathbb {E}}(e^{t (Z_{1}-2)})=\frac {e^{-2t}}{1-2t}$ for t ∈ (0, 1/2) and ${\mathbb {E}}(e^{-t^{\prime }(Z_{1}-2)})=\frac {e^{2 t^{\prime }}}{1 + 2t^{\prime }}$ for all $t^{\prime }\geq 0$, after the minimization we obtain

$$ {\mathbb{P}}(|Y_{K}|\geq \alpha\ell^{1/3}) \leq 2 e^{-\alpha^{2}/(16 K)(1+\mathcal{O}(\alpha K^{-1} \ell^{-1/3})}\leq 4 e^{-\alpha^{2}/(16 K)} $$

(C.7)

for all ℓ large enough.

For the second estimate, from the inequality

$$\begin{array}{@{}rcl@{}} &&{\mathbb{P}}(\max\limits_{u\geq K}{L^{\mathcal{B}}}_{(0,0)\to I(u)} -{L^{\mathcal{B}}}_{(0,0)\to I(K)}-\beta u^{2} \ell^{1/3}\geq \alpha \ell^{1/3})\\ &\leq &\sum\limits_{m\geq 1}{\mathbb{P}}(\max\limits_{u\in [K m,K(m + 1)]}{L^{\mathcal{B}}}_{(0,0)\to I(u)} -{L^{\mathcal{B}}}_{(0,0)\to I(K)}\geq (\alpha+\beta K^{2} m^{2}) \ell^{1/3})\\ &\leq & \sum\limits_{m\geq 1} \inf\limits_{t>0} e^{-t(\alpha+\beta K^{2} m^{2})\ell^{1/3}} {\mathbb{E}}(e^{t(Z_{1}-2)})^{2K m \ell^{2/3}}. \end{array} $$

(C.8)

Maximising over t and taking the sum we finally get¹

$$ {\mathbb{P}}\left( \max_{u\geq K}{L^{\mathcal{B}}}_{(0,0)\to I(u)} -{L^{\mathcal{B}}}_{(0,0)\to I(-K)}-\beta u^{2} \ell^{1/3}\geq \alpha \ell^{1/3}\right) \leq C e^{-\frac{(\alpha+\beta K^{2})^{2}}{16 K}} $$

(C.9)

for a constant C and for all β > 0 and α > −βK² and ℓ large enough. □

Appendix E: Bounds for point-to-half line LPP

Proposition E.1

Let I(u) = (τN,τN) + u(2N)^2/3(1,− 1). Then,

$$ {\mathbb{P}}\left( \max_{|u|>M} L_{I(u)\to (N,N)}>4(1-\tau)N + 2^{4/3}(s-\gamma M^{2}) N^{1/3}\right)\leq C e^{-c M^{2} (1-\tau)^{-4/3}} e^{-\tilde c s (1-\tau)^{-1/3}} $$

(E.1)

for some constants $C,c,\tilde c>0$, which can be taken uniform in N and uniform for γ in a compact subset of (0, 1/(1 − τ)).

Proof

By symmetry, it is enough to get the bound on the distribution of $\max _{u<-M} L_{I(u)\to (N,N)}$. By first shifting I(−M) to the origin, and then using the mapping between LPP and TASEP, the distribution function is the same as the distribution of TASEP particle number $n=t/4+\tilde \tau (t/2)^{2/3}$ at time t = 4(1 − τ)N + 2^4/3N^1/3(s − γM²), starting at x_k(0) = − 2k,k ≥ 0.

From Proposition 3 of [12] we have an explicit expression in terms of Fredholm determinant. The upper tail estimate is standard. Using Hadamard’s bound it is enough to have a bound on the correlation kernel. In Section 4 of [12] exponential decay of the rescaled correlation kernel has been proven. Then, simple algebraic computations give the claimed result.