Probabilistic and Geometric Methods in Last Passage Percolation
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Probabilistic and geometric methods in last passage percolation by Milind Hegde A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Alan Hammond, Co‐chair Professor Shirshendu Ganguly, Co‐chair Professor Fraydoun Rezakhanlou Professor Alistair Sinclair Spring 2021 Probabilistic and geometric methods in last passage percolation Copyright 2021 by Milind Hegde 1 Abstract Probabilistic and geometric methods in last passage percolation by Milind Hegde Doctor of Philosophy in Mathematics University of California, Berkeley Professor Alan Hammond, Co‐chair Professor Shirshendu Ganguly, Co‐chair 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 as‐ signed 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 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 prop‐ erties, 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 cer‐ tain 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 Airy2 process P1 and the parabolic Airy line ensemble P 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 P1, 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 al‐ lowed to vary along a line. The ensemble P possess a probabilistic resampling property known as the Brownian Gibbs property, which gives an explicit description of a certain conditional 2 distribution of P in terms of non‐intersecting Brownian bridges. This property gives a qualita‐ tive comparison—absolute continuity—of P1 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 " under the law of Brownian motion has probability 1−o(1) −1 5/6 at most " under the law of an increment of P1, with an explicit form of exp(O(1)(log " ) ) for the "−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 P1 with respect to Brownian motion lies in all Lp spaces with p 2 [1; 1), i.e., has finite polynomial moments of all orders. The bounds also hold for lower curves of P 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 Z2 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 i.i.d. non‐negative random variables associated to the vertices of Z2, 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 pas‐ sage value Xr from (1; 1) to (r; r). The GUE Tracy‐Widom distribution is well‐known to have 3 upper and lower tail exponents of 2 and 3, which is also known for the prelimiting Xr in the mentioned exactly solvable models. Here we work in a more general setup and adopt some nat‐ ural 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 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) 7! ht(x) is a space‐time process constructed in [MQR17] which should be thought of as a limiting object analogous to P1 under general initial data. We show that the event, for fixed t > 0, that ht has "‐twin peaks, i.e., there exists a point at a given distance away from the maximizer of ht at which ht comes within " of its maximum, has probability at least of order ". This served as an important input in [CHHM21] to prove that the Hausdorff dimension of the set of exceptional 2 times t where ht has multiple maximizers is almost surely 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. i To family, friends, mentors, and the simple things which suffice. ii Contents Contents ii List of Figures v I Introduction 1 1 The KPZ universality class 2 1.1 The Eden model and first passage percolation .................. 2 1.2 Integrability within KPZ .............................. 5 1.3 Last passage percolation .............................. 9 1.4 The KPZ equation and the KPZ fixed point .................... 17 2 Probabilistic and geometric viewpoints 21 2.1 Background on the probabilistic approach .................... 22 2.2 Results in the probabilistic stream: Brownian regularity ............. 28 2.3 Background on the geometric approach ...................... 31 2.4 Results in the geometric stream: Bootstrapping .................. 34 2.5 Results combining the perspectives: Fractal geometry .............. 36 2.6 Other graduate work ............................... 40 II Brownian regularity 43 3 Proof context 44 3.1 Pertinent recent work ............................... 44 3.2 Method of proof .................................. 47 3.3 Organization of Part II .............................. 49 4 Notation and setup 50 4.1 Notation, Brownian Gibbs, and regular ensembles ................ 50 4.2 An important example of regular ensembles: Brownian LPP weight profiles . 53 4.3 Main result .................................... 55 iii 5 Proof framework 57 5.1 The jump ensemble ................................ 57 5.2 A conceptual framework in terms of costs ..................... 76 6 Proving the density bounds 89 6.1 The easy case: Below the pole on both sides ................... 89 6.2 The moderate case: Above the pole on both sides ................ 91 6.3 The difficult case: Above and below the pole on either side ........... 102 6.4 When no pole is present .............................. 110 III Bootstrapping 114 7 The basic idea of bootstrapping 115 7.1 Introduction, the model, and assumptions .................... 115 7.2 Main results .................................... 121 7.3 The key ideas ................................... 125 7.4 Tails for constrained weight ............................ 131 7.5 Related work .................................... 132 7.6 A few important tools ............................... 133 7.7 Organization of Part III .............................. 135 8 Concentration tools and the bootstrap 136 9 The tail bound proofs 140 9.1 Upper tail bounds ................................. 140 9.2 Lower bound on upper tail ............................ 151 9.3 Lower tail and constrained lower tail bounds ................... 152 10 Constructing disjoint high weight paths 162 10.1 The construction in outline ............................ 163 10.2 The construction in detail ............................. 164 10.3 Bounding below the expected weight of the construction ............. 168 10.4 One‐sided concentration of the construction weight ............... 170 IV Twinpeaks for the KPZ fixed point 173 11 The lower bound on the twin peaks probability 174 11.1 Preliminaries .................................... 176 11.2 Locations of maximizers .............................. 181 11.3 The resampling framework ............................ 184 11.4 The Brownian Gibbs property ..........................