Optimal tail exponents in general last passage percolation via bootstrapping & geodesic geometry

Abstract

We consider last passage percolation on $\mathbb Z^2$ with general weight distributions, which is expected to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. In this model, an oriented path between given endpoints which maximizes the sum of the i.i.d. weight variables associated to its vertices is called a geodesic. Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the assumptions to prove optimal upper and lower tail behavior with the exponents of $3/2$ and $3$ for the weight of the geodesic from $(1,1)$ to $(r,r)$ for all large finite $r$. The proofs merge several ideas, including the well known super-additivity property of last passage values, concentration of measure behavior for sums of stretched exponential random variables, and geometric insights coming from the study of geodesics and more general objects called geodesic watermelons. Previously such optimal behavior was only known for exactly solvable models, with proofs relying on hard analysis of formulas from integrable probability, which are unavailable in the general setting. Our results illustrate a facet of universality in a class of KPZ stochastic growth models and provide a geometric explanation of the upper and lower tail exponents of the GUE Tracy-Widom distribution, the conjectured one point scaling limit of such models. The key arguments are based on an observation of general interest that super-additivity allows a natural iterative bootstrapping procedure to obtain improved tail estimates.

Figures (7)

• Figure 1: In green is depicted the heaviest path which passes through the selection of intervals in blue. The cyan curve between the second and third (similarly the third and fourth) blue intervals is the heaviest path with endpoints on those intervals. Because these consecutive cyan paths do not need to share endpoints, the weight of the green path is at most the sum of the interval-to-interval weights defined by the blue intervals, which provides the substitute sub-additive relation.
• Figure 2: A simulation of the $k$-geodesic watermelon in the related model of Poissonian last passage percolation for $k=10$.
• Figure 3: The grid of $k^2$ intervals for the lower bound of the lower tail. An interval and the following row of intervals are blue: consider the event that the heaviest path from the former to the latter is at most $\mu r/k - C(r/k)^{1/3}$. To prove that this has positive probability, we make use of parabolic curvature of the weight profile (shown in green) to argue that if the endpoint on the row is too extreme, it will typically suffer the loss we want; a separate backing up argument is employed for when the endpoint is near the center where the parabolic weight loss is not significant.
• Figure 4: The grid utilized for the discretization in Step 1 of the proof of Proposition \ref{['p.upper tail bootstrap iteration']}. Note that measurements are made along the antidiagonal and diagonal only, with the diagonal chosen over the line with the slope of the left or right boundary of the grid. The lower boundary of the grid $\mathbb G^z$ is centered at $(1,1)$ and the upper boundary at $(r-z,r+z)$. From each grid line $\mathbb G_i^z$, one interval $L_i$ is picked to form a discretization $\mathcal{L}^z = (L_0, \ldots, L_k)$ with the constraint that $L_0$ is fixed to be the interval on $\mathbb G_0^z$ whose midpoint is $(1,1)$ and $L_k$ to be the interval on $\mathbb G_k^z$ whose midpoint is $(r-z,r+z)$. On the high probability event that all geodesics passes through the grid, its weight is upper bounded by the maximum, over all discretizations $\mathcal{L}^z$, of the sum of interval-to-interval weights of the intervals in $\mathcal{L}^z$. These weights are independent and have fluctuations of scale $(r/k)^{1/3}$, which allows us to use the idea of bootstrapping.
• Figure 5: The argument for Lemma \ref{['l.Z tail bound']}. The two black intervals have midpoint separation of $z$ in the antidiagonal direction. The orange path (dashed) is the heaviest path between the two intervals (so has weight $Z$), and the brown paths (solid) are geodesics connecting the black points to the endpoints of the red path. The green path (dotted) is a geodesic between the two black points. With positive probability the two brown paths each have weight greater than $\mu \delta_j r - \frac{1}{3}\theta r^{1/3}$, and so, on the intersection of those events with $\{Z > \mu r - \lambda_{j+1}Gz^2/r + \theta r^{1/3}\}$, it holds that the green path has weight at least $\mu(1+2\delta_j)r - \lambda_{j+1}Gz^2/r + \frac{1}{3}\theta r^{1/3}$. We choose $\delta_j$ such that the parabolic term in this expression is $\lambda_j Gz^2/(1+2\delta_j)r$ and apply the point-to-point bound we have. It is because the antidiagonal separation between each pair of black and green points is zero that we have a decrease in the parabolic term. If we instead make this separation proportional to $z$, then there is no decrease in the parabolic term, but for large $z$ the gradient of the limit shape from Assumption \ref{['a.limit shape assumption']} causes issues. This can be more carefully handled if we instead consider the supremum of fluctuations of point-to-point weights from their expectation, and we will have need to do this on one occasion in the appendix. We also note an inaccuracy in the figure which we have retained to not distract from the main point: in truth, the brown and green paths will have some amount of overlap around their starting and ending points, as a general phenomenon of geodesic coalescence.

Theorems & Definitions (52)

• Theorem 1.1: Theorem 1.6 of johansson2000shape
• Theorem 1.2: seppalainen1998couplingjohansson2000shapeledoux2010basu2019lower
• Remark 1.3
• Theorem 1: Upper-tail upper bound
• Theorem 2: Upper-tail lower bound
• Theorem 3: Lower-tail upper bound
• Theorem 4: Lower-tail lower bound
• Remark 1.4: Relation of tail exponents to fluctuation exponents
• Remark 1.5: Suboptimal log factor in Theorem \ref{['t.upper tail bootstrapping']}
• Remark 1.6: $\zeta(\alpha)\to 0$ as $\alpha\to 0$