Milind Hegde

Bio: Milind Hegde is an academic researcher from University of California, Berkeley. The author has contributed to research in topics: Geodesic & Law of the iterated logarithm. The author has an hindex of 6, co-authored 9 publications receiving 82 citations.

Brownian structure in the KPZ fixed point

Abstract: Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds. When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be embedded via the Robinson-Schensted-Knuth correspondence as the uppermost curve in an $\mathbb{N}$-indexed system of random continuous curves, the Airy line ensemble. Among our principal results is the assertion that the Airy$_2$ process enjoys a very strong similarity to Brownian motion $B$ (of rate two) on unit-order intervals; as a consequence, the Radon-Nikodym derivative of the law of $\mathcal{A}$ on say $[-1,1]$, with respect to the law of $B$ on this interval, lies in every $L^p$ space for $p \in (1,\infty)$. Our technique of proof harnesses a probabilistic resampling or {\em Brownian Gibbs} property satisfied by the Airy line ensemble after parabolic shift, and this article develops Brownian Gibbs analysis of this ensemble begun in [CH14] and pursued in [Ham19a]. Our Brownian comparison for scaled interface profiles is an element in the ongoing programme of studying KPZ universality via probabilistic and geometric methods of proof, aided by limited but essential use of integrable inputs. Indeed, the comparison result is a useful tool for studying this universality class. We present and prove several applications, concerning for example the structure of near ground states in Brownian last passage percolation, or Brownian structure in scaled interface profiles that arise from evolution from any element in a very general class of initial data.

Interlacing and scaling exponents for the geodesic watermelon in last passage percolation

Abstract: In discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $\mathbb Z^2$, and each finite upright path in $\mathbb Z^2$ is ascribed the weight given by the sum of values of its vertices The weight of a collection of disjoint paths is the sum of its members' weights The notion of a geodesic, a maximum weight path between two vertices, has a natural generalization concerning several disjoint paths: a $k$-geodesic watermelon in $[1,n]^2\cap\mathbb Z^2$ is a collection of $k$ disjoint paths contained in this square that has maximum weight among all such collections While the weights of such collections are known to be important objects, the maximizing paths have been largely unexplored beyond the $k=1$ case For exactly solvable models, such as exponential and geometric LPP, it is well known that for $k=1$ the exponents that govern fluctuation in weight and transversal distance are $1/3$ and $2/3$; that is, typically, the weight of the geodesic on the route $(1,1) \to (n,n)$ fluctuates around a dominant linear growth of the form $\mu n$ by the order of $n^{1/3}$; and the maximum Euclidean distance of the geodesic from the diagonal has order $n^{2/3}$ Assuming a strong but local form of convexity and one-point moderate deviation bounds for the geodesic weight profile---which are available in all known exactly solvable models---we establish that, typically, the $k$-geodesic watermelon's weight falls below $\mu nk$ by order $k^{5/3}n^{1/3}$, and its transversal fluctuation is of order $k^{1/3}n^{2/3}$ Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit Our methods also yield sharp rigidity estimates for naturally associated point processes, which improve on estimates obtained via tools from the theory of determinantal point processes available in the integrable setting

Lower Deviations in $\beta$-ensembles and Law of Iterated Logarithm in Last Passage Percolation

Abstract: For the last passage percolation (LPP) on $\mathbb{Z}^2$ with exponential passage times, let $T_{n}$ denote the passage time from $(1,1)$ to $(n,n)$. We investigate the law of iterated logarithm of the sequence $\{T_{n}\}_{n\geq 1}$; we show that $\liminf_{n\to \infty} \frac{T_{n}-4n}{n^{1/3}(\log \log n)^{1/3}}$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (J. Theor. Probab., 2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight shift in perspective from point-to-point passage times to considering point-to-line passage times instead, and exploiting the correspondence of the latter to the largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical ingredient, which is of independent interest, is a new lower bound of lower tail deviation probability of the largest eigenvalue of $\beta$-Laguerre ensembles, which extends the results proved in the context of the $\beta$-Hermite ensembles by Ledoux and Rider (Electron. J. Probab., 2010).

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 iid 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

Local and global comparisons of the Airy difference profile to Brownian local time

Abstract: There has recently been much activity within the Kardar-Parisi-Zhang universality class spurred by the construction of the canonical limiting object, the parabolic Airy sheet $\mathcal{S}:\mathbb{R}^2\to\mathbb{R}$ [arXiv:1812.00309]. The parabolic Airy sheet provides a coupling of parabolic Airy$_2$ processes -- a universal limiting geodesic weight profile in planar last passage percolation models -- and a natural goal is to understand this coupling. Geodesic geometry suggests that the difference of two parabolic Airy$_2$ processes, i.e., a difference profile, encodes important structural information. This difference profile $\mathcal{D}$, given by $\mathbb{R}\to\mathbb{R}:x\mapsto \mathcal{S}(1,x)- \mathcal{S}(-1,x)$, was first studied by Basu, Ganguly, and Hammond [arXiv:1904.01717], who showed that it is monotone and almost everywhere constant, with its points of non-constancy forming a set of Hausdorff dimension $1/2$. Noticing that this is also the Hausdorff dimension of the zero set of Brownian motion leads to the question: is there a connection between $\mathcal{D}$ and Brownian local time? Establishing that there is indeed a connection, we prove two results. On a global scale, we show that $\mathcal{D}$ can be written as a Brownian local time patchwork quilt, i.e., as a concatenation of random restrictions of functions which are each absolutely continuous to Brownian local time (of rate four) away from the origin. On a local scale, we explicitly obtain Brownian local time of rate four as a local limit of $\mathcal{D}$ at a point of increase, picked by a number of methods, including at a typical point sampled according to the distribution function $\mathcal{D}$. Our arguments rely on the representation of $\mathcal{S}$ in terms of a last passage problem through the parabolic Airy line ensemble and an understanding of geodesic geometry at deterministic and random times.

Shape Fluctuations and Random Matrices

Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

The Stochastic Process π

Abstract: The principal aim of this chapter is to familiarize the reader with the fact that the conditional distribution of the signal can be viewed as a stochastic process with values in the space of probability measures. While it is true that this chapter sets the scene for the subsequent chapters, it can be skipped by those readers whose interests are biased towards the applied aspects of the subject. The gist of the chapter can be summarized by the following.

GUEs and queues

Abstract: Consider the process D k , k = 1,2,…, given by B i being independent standard Brownian motions. This process describes the limiting behavior “near the edge” in queues in series, totally asymmetric exclusion processes or oriented percolation. The problem of finding the distribution of D. was posed in [GW]. The main result of this paper is that the process D. has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from Gaussian Unitary Ensemble.

Bulk properties of the Airy line ensemble

Abstract: The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson's Brownian motion converging to the Airy line ensemble.