Probabilistic and geometric methods in last passage percolation

Last passage percolation (LPP) refers to a broad class of models thought to lie within the Kardar-Parisi-Zhang universality class of one-dimensional stochastic growth models. In LPP models, there is a planar random noise environment through which directed paths travel; paths are assigned a weight based on their journey through the environment, usually by, in some sense, integrating the noise over the path. For given points $y$ and $x$, the weight from $y$ to $x$ is defined by maximizing the weight over all paths from $y$ to $x$. A path which achieves the maximum weight is called a \emph{geodesic}.A few last passage percolation models are exactly solvable, i.e., possess what is called integrable structure. This gives rise to valuable information, such as explicit probabilistic resampling properties, distributional convergence information, or one point tail bounds of the weight profile as the starting point $y$ is fixed and the ending point $x$ varies. However, much of the behaviour currently proved within only the special models with integrable structure is expected to hold in a much larger class of models; further, integrable techniques alone seem unable to obtain certain types of information, such as process-level properties of natural limiting objects.% This thesis explores probabilistic and geometric approaches which assume limited integrable input to infer information unavailable via exactly solvable techniques or to develop methods which will prove be to robust and eventually applicable to a broader class of non-integrable models.In the first part, we study the parabolic Airy$_2$ process $\cP_1$ and the parabolic Airy line ensemble $\cP$ via a probabilistic resampling structure enjoyed by the latter. The parabolic Airy line ensemble is an infinite $\N$-indexed collection of random continuous non-intersecting curves, whose top (and lowest indexed) curve is $\cP_1$, which is a limiting weight profile in LPP when the starting point is fixed (which can be thought of as a particular form of initial data) and the ending point is allowed to vary along a line. The ensemble $\cP$ possess a probabilistic resampling property known as the Brownian Gibbs property, which gives an explicit description of a certain conditional distribution of $\cP$ in terms of non-intersecting Brownian bridges. This property gives a qualitative comparison---absolute continuity---of $\cP_1$ to Brownian motion, as proved in [CH14]. Using a framework introduced in [Ham19a], we prove here a strong quantitative form of comparison, showing that an event which has probability $\varepsilon$ under the law of Brownian motion has probability at most $\varepsilon^{1-o(1)}$ under the law of an increment of $\cP_1$, with an explicit form of $\exp(O(1)(\log\varepsilon^{-1})^{5/6})$ for the $\varepsilon^{-o(1)}$ error factor. Up to the error factor, this is expected to be sharp. One consequence is that the Radon-Nikodym derivative of the increment of $\cP_1$ with respect to Brownian motion lies in all $L^p$ spaces with $p\in[1,\infty)$, i.e., has finite polynomial moments of all orders. The bounds also hold for lower curves of $\cP$ and for weight profiles in an LPP model known as Brownian LPP. In the second part, we work in an LPP model on the lattice $\Z^2$ and make use of a geometric approach, i.e., we study properties of geodesics and other weight maximizing structures. The random environment is given by \iid non-negative random variables associated to the vertices of $\Z^2$, and the weight of an up-right path is the sum of the random variables it passes through. Now, in exactly solvable models such as when the vertex weight distribution is geometric or exponential, the GUE Tracy-Widom distribution is known to be a scaling limit of the last passage value $X_r$ from $(1,1)$ to $(r,r)$. The GUE Tracy-Widom distribution is well-known to have upper and lower tail exponents of $\frac{3}{2}$ and $3$, which is also known for the prelimiting $X_r$ in the mentioned exactly solvable models. Here we work in a more general setup and adopt some natural assumptions of curvature and weak one-point upper and lower tail estimates on the weight profile---with no assumptions on the vertex weight distribution and, hence, a non-integrable setting---which we bootstrap up to obtain the optimal tail exponents for both tails, in terms of both upper and lower bounds. We also obtain sharp upper and lower bounds for the lower tail of the maximum weight over all paths which are \emph{constrained} to lie in a strip of given width, a result which was not previously known in even the integrable models and does not seem easily accessible via integrable techniques.Finally, in the third part, we combine the geometric and probabilistic approaches to obtain an estimate on a certain probability of the KPZ fixed point. The KPZ fixed point $(t,x)\mapsto \mf h_t(x)$ is a space-time process constructed in [MQR17] which should be thought of as a limiting object analogous to $\cP_1$ under general initial data. We show that the event, for fixed $t>0$, that $\mf h_t$ has \emph{$\varepsilon$-twin peaks}, i.e., there exists a point at a given distance away from the maximizer of $\mf h_t$ at which $\mf h_t$ comes within $\varepsilon$ of its maximum, has probability at least of order $\varepsilon$. This served as an important input in [CHHM21] to prove that the Hausdorff dimension of the set of exceptional times $t$ where $\mf h_t$ has multiple maximizers is almost surely $\frac{2}{3}$, on the positive-probability event that the set is non-empty. Geometric and probabilistic arguments are combined by making use of a remarkable identity of last passage values between an original and a certain transformed environment proven in [DOV18], which allows us to consider geodesics through an environment which itself has the Brownian Gibbs property