arXiv subjects
Publications and source records attributed to Riddhipratim Basu.
We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi,\omega)$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.
For first passage percolation (FPP) on Euclidean lattices $\mathbb{Z}^d$ with $d\ge 2$, it is expected that the variance of the first passage time between two points grows sublinearly in the distance with a universal exponent strictly smaller than $1$. Following Kesten's $O(n)$ upper bound (Ann. Appl. Probab., 1993) on the variance, Benjamini, Kalai and Schramm (Ann. Probab., 2003) used hypercontractivity to obtain an improvement of a factor of $\log n$ when passage times take two values with equal probability. This was later extended to more general classes of passage time distributions. However, unlike in exactly solvable planar models in last passage percolation where the variance is known to be $\Theta(n^{2/3})$, the best known upper bound for the variance of passage times has remained $O(n/\log n)$ in all non-trivial variants of FPP. For a class of rotationally invariant Riemannian FPP on the plane, we show that the variance is $O(n^{1-\varepsilon})$ for some $\varepsilon>0$. Our argument uses fluctuation estimates for passage times and geodesics derived in Basu, Sidoravicius and Sly (2023) together with a multi-scale argument to establish that the geodesic exhibits disorder chaos, i.e., upon resampling a small fraction of the underlying randomness, the updated geodesic has on average a small overlap with the original one; this, established at a large number of scales, leads to a polynomial improvement of the variance bound.
The study of transversal fluctuations of the optimal path is a crucial aspect of the Kardar-Parisi-Zhang (KPZ) universality class. In this work, we establish the large deviation limit for the midpoint transversal fluctuations in a general last-passage percolation (LPP) model with mild assumption on the i.i.d. weights. The rate function is expressed in terms of the right tail large deviation rate function of the last-passage value and the shape function. When the weights are chosen to be i.i.d. exponential random variables, our result verifies a conjecture communicated to us by Liu [Liu'22], showing the asymptotic probability of the geodesic from $(0,0)$ to $(n,n)$ following the corner path $(0,0) \to (n,0) \to (n,n)$ is $({4}/{e^2})^{n+o(n)}$.
We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that, given any fixed direction $\xi\in \partial G$, and under mild conditions on the passage time distribution, there exists almost surely a unique semi-infinite FPP geodesic from each $v\in G$ to $\xi$. Also, these geodesics coalesce to form a tree. Our main topic of study is the set of (random) exceptional directions for which uniqueness or coalescence fails. We study these directions in the context of two random geodesics trees: one formed by the union of all geodesics starting at a given base point, and the other formed by the union of all semi-infinite geodesics in a given direction $\xi\in \partial G$. We show that, under mild conditions, the set of exceptional directions almost surely has a strictly smaller Hausdorff dimension than the boundary, and hence has measure zero with respect to the Patterson-Sullivan measure. We also establish an upper bound on the maximum number of disjoint geodesics in the same direction. For groups that are not virtually free, we show that almost surely exceptional directions exist and are dense in $\partial G$. When the topological dimension of $\partial G$ is greater than one, we establish the existence of uncountably many exceptional directions. When the topological dimension of $\partial G$ is $n$, we prove the existence of directions $\xi$ with at least $(n+1)$ disjoint geodesics. Our results hinge on deep facts about hyperbolic groups. En route, we also establish facts about the structure of random bigeodesics that substantially strengthen prior results.
It is well-known that the largest eigenvalue of an $n\times n$ GUE matrix and the length of a longest increasing subsequence in a uniform random permutation of length $n$, both converge weakly to the GUE Tracy-Widom distribution as $n\to \infty$. We consider the sequences of the largest eigenvalues of the $n\times n$ principal minor of an infinite GUE matrix, and the the lengths of longest increasing subsequences of a growing sequence of random permutations (which, by the RSK bijection corresponds to the top row of the Young diagrams growing according to the Plancherel growth process), and establish laws of fractional logarithms for these. That is, we show that, under a further scaling of $(\log n)^{2/3}$ and $(\log n)^{1/3}$, the $\limsup$ and $\liminf$ respectively of these scaled quantities converge almost surely to explicit non-zero and finite constants. Our results provide complete solutions to two questions raised by Kalai in 2013. We affirm a conjecture of Paquette and Zeitouni (Ann. Probab., 2017), and give a new proof of $\limsup$, due to Paquette and Zeitouni (Ann. Probab., 2017), who provided a partial solution in the case of GUE minor process.
We establish limit theorems for the maxima and minima of Airy$_1$ and Airy$_2$ processes (denoted by $\mathcal{A}_1(\cdot)$ and $\mathcal{A}_2(\cdot)$ respectively) over growing intervals. In particular, we identify the finite non-zero constants that are the almost sure limits of $(\log t)^{-2/3}\max_{0\le s \le t} \mathcal{A}_{i}(s)$ and $(\log t)^{-1/3}\min_{0\le s \le t} \mathcal{A}_{i}(s)$ for $i=1,2$. This complements and extends the results of (Pu, 2023), where the question for the maxima was considered and the order of growth was identified for both $\mathcal{A}_1$ and $\mathcal{A}_2$ and the constant was identified for $\mathcal{A}_1$. Our approach is different from that of (Pu, 2023); instead of complicated formulae for multi-point distributions, we rely on the well-known convergence of passage time profiles in planar exponential last passage percolation started from different initial conditions to $\mathcal{A}_1$ and $\mathcal{A}_2$, together with the recently developed sharp one-point estimates in (Baslingker et al., 2024) for the point-to-point and point-to-line passage times in exponential LPP and a combination of old and new results on the geometry of the LPP landscape.
For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point geodesic exhibits an atypically large transversal fluctuation at the halfway point. In particular, we show that $p^*_n(t)$, the probability that the point-to-line geodesic from the origin to the line $x+y=2n$ ends at $(n-t(2n)^{2/3}, n+t(2n)^{2/3})$ satisfies that $n^{2/3}p^*_n(t)=\exp(-(\frac{4}{3}+o(1))t^{3})$ for $t$ large and $p_{n,\frac{1}{2}}(t)$, the probability that the geodesic from the origin to the point $(n,n)$ passes through the point $(\frac{1}{2}n-tn^{2/3}, \frac{1}{2} n+tn^{2/3})$, satisfies $n^{2/3}p_{n,\frac{1}{2}}(t)=\exp(-(\frac{8}{3}+o(1))t^3)$ for $t$ large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022).
Hermite and Laguerre $\beta$-ensembles are important and well studied models in random matrix theory with special cases $\beta=1,2,4$ corresponding to eigenvalues of classical random matrix ensembles. It is well known that the largest eigenvalues in these, under appropriate scaling, converge weakly to the Tracy-Widom $\beta$ distribution whose distribution function $F_{\beta}$ has asymptotics given by $1-F_{\beta}(x)=\exp\left(-\frac{2\beta}{3}(1+o(1))x^{3/2}\right)$ as $x\to \infty$ and $F_{\beta}(x)=\exp\left(-\frac{\beta}{24}(1+o(1))|x|^3\right)$ as $x\to -\infty$. Although tail estimates for the largest eigenvalues with correct exponents have been proved for the pre-limiting models, estimates with matching constants had not so far been established for general $\beta$; even in the exactly solvable cases, some of the bounds were missing. In this paper, we prove upper and lower moderate deviation estimates for both tails with matching constants. We illustrate the usefulness of these estimates by considering certain questions in planar exponential last passage percolation (LPP), a well-studied model in the KPZ universality class in which certain statistics have same distributions as largest eigenvalues in Laguerre $\beta$-ensembles (for $\beta=1,2,4$). Using our estimates in conjunction with a combination of old and new results on the LPP geometry, we obtain three laws of iterated logarithm including one which settles a conjecture of Ledoux (J. Theor. Probab., 2018). We expect that the sharp moderate deviation estimates will find many further applications in LPP problems and beyond.
For rotationally invariant first passage percolation (FPP) on the plane, we use a multi-scale argument to prove stretched exponential concentration of the first passage times at the scale of the standard deviation. Our results are proved under hypotheses which can be verified for many standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the Howard-Newman model. This is the first such tight concentration result known for any model that is not exactly solvable. As a consequence, we prove a version of the so called KPZ relation between the passage time fluctuations and the transversal fluctuations of geodesics as well as up to constant upper and lower bounds for the non-random fluctuations in these models. Similar results have previously been known conditionally under unproven hypotheses, but our results are the first ones that apply to some specific FPP models. Our arguments are expected to be useful in proving a number of other estimates which were hitherto only known conditionally or for exactly solvable models.
For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, it is known that given any non-axial direction, all the semi-infinite geodesics starting from points in $\mathbb{Z}^2$ in that direction almost surely coalesce, thereby forming a geodesic tree which has only one end. It is widely understood that the geodesic trees are important objects in understanding the geometry of the LPP landscape. In this paper we study several natural questions about these geodesic trees and their intersections. In particular, we obtain optimal (up to constants) upper and lower bounds for the (power law) tails of the height and the volume of the backward sub-tree rooted at a fixed point. We also obtain bounds for the probability that the sub-tree contains a specific vertex, e.g. the sub-tree in the direction $(1,1)$ rooted at the origin contains the vertex $-(n,n)$, which answers a question analogous to the well-known midpoint problem in the context of semi-infinite geodesics. Furthermore, we obtain bounds for the probability that a pair of intersecting geodesics both pass through a given vertex. These results are interesting in their own right as well as useful in several other applications.
Temporal correlation for randomly growing interfaces in the KPZ universality class is a topic of recent interest. Most of the works so far have been concentrated on the zero temperature model of exponential last passage percolation, and three special initial conditions, namely droplet, flat and stationary. We focus on studying the time correlation problem for generic random initial conditions with diffusive growth. We formulate our results in terms of the positive temperature exactly solvable model of the inverse-gamma polymer and obtain up to constant upper and lower bounds for the correlation between the free energy of two polymers whose endpoints are close together or far apart. Our proofs apply almost verbatim to the zero temperature set-up of exponential LPP and are valid for a broad class of initial conditions. Our work complements and completes the partial results obtained in (Ferrari-Occelli'19), following the conjectures of (Ferrari-Spohn'16). Moreover, our arguments rely on the one-point moderate deviation estimates which have recently been obtained using stationary polymer techniques and thus do not depend on complicated exact formulae.
Understanding the decay of correlations in time for (1+1)-dimensional polymer models in the KPZ universality class has been a challenging topic. Following numerical studies by physicists, concrete conjectures were formulated by Ferrari and Spohn (Ferrari-Spohn '16) in the context of planar exponential last passage percolation. These have mostly been resolved by various authors. In the context of positive temperature lattice models, however, these questions have remained open. We consider the time correlation problem for the exactly solvable inverse-gamma polymer in $\mathbb{Z}^2$. We establish, up to constant factors, upper and lower bounds on the correlation between free energy functions for two polymers rooted at the origin (droplet initial condition) when the endpoints are either close together or far apart. We find the same exponents as predicted in (Ferrari-Spohn '16). Our arguments rely on the understanding of stationary polymers, coupling, and random walk comparison. We use recently established moderate deviation estimates for the free energy. In particular, we do not require asymptotic analysis of complicated exact formulae.
For the directed landscape, the putative universal space-time scaling limit object in the (1+1) dimensional Kardar-Parisi-Zhang (KPZ) universality class, consider the geodesic tree -- the tree formed by the coalescing semi-infinite geodesics in a given direction. As shown in Bhatia '23, this tree comes interlocked with a dual tree, which (up to a reflection) has the same marginal law as the geodesic tree. Analogous examples of one ended planar trees formed by coalescent semi-infinite random paths and their duals are objects of interest in various other probability models, a classical example being the Brownian web, which is constructed as a scaling limit of coalescent random walks. In this paper, we continue the study of the geodesic tree and its dual in the directed landscape and exhibit a new space-filling curve traversing between the two trees that is naturally parametrized by the area it covers and encodes the geometry of the two trees; this parallels the construction of the T\'oth-Werner curve between the Brownian web and its dual. We study the regularity and fractal properties of this Peano curve, exploiting simultaneously the symmetries of the directed landscape and probabilistic estimates obtained in planar exponential last passage percolation, which is known to converge to the directed landscape in the scaling limit. On the way, we develop a novel coalescence estimate for geodesics, and this has recently found application in other work.
We study the decay of the covariance of the Airy$_1$ process, $\mathcal{A}_1$, a stationary stochastic process on $\mathbb{R}$ that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $\textrm{Cov}(\mathcal{A}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$ as $u\to\infty$. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $\mathcal{A}_1$ to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the Airy$_1$ process together with the FKG inequality.
First passage percolation (FPP) on $\mathbb{Z}^d$ or $\mathbb{R}^d$ is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesic and the typical one. We investigate this in the context of $γ$-Liouville Quantum Gravity (LQG) (where $γ\in (0,2)$ is a parameter) -- a random Riemannian surface induced on the complex plane by the random metric tensor $e^{2γh/d_γ} ({dx^2+dy^2}),$ where $h$ is the whole plane, properly centered, Gaussian Free Field (GFF), and $d_γ$ is the associated dimension. We consider the unique infinite geodesic $Γ$ from the origin, parametrized by the logarithm of its chemical length, and show that, for an almost sure realization of $h$, the distributions of the appropriately scaled field and the induced metric on a ball, rooted at a point "uniformly" sampled on $Γ$, converge to deterministic measures on the space of generalized functions and continuous metrics on the unit disk respectively. Moreover, we show that the limiting objects living on the unit disk are singular with respect to their typical counterparts, but become absolutely continuous away from the origin. Our arguments rely on unearthing a regeneration structure with fast decay of correlation in the geodesic owing to coalescence and the domain Markov property of the GFF. While there have been significant recent advances around this question for stochastic planar growth models in the KPZ class, the present work initiates this research program in the context of LQG.
Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics, however the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on $\mathbb{Z}^2$ with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integrable probability literature to study coalescence structure of finite geodesics, thereby making rigorous a heuristic argument due to Newman.
We consider directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn (SIGMA, 2016), we show that for the passage times from the line $x+y=0$ to the points $(r,r)$ and $(n,n)$, denoted $X_{r}$ and $X_{n}$ respectively, as $n\to \infty$ and $\frac{r}{n}$ is small but bounded away from zero, the covariance satisfies $$\mbox{Cov}(X_{r},X_{n})=Θ\left((\frac{r}{n})^{4/3+o(1)} n^{2/3}\right),$$ thereby establishing $\frac{4}{3}$ as the temporal covariance exponent. This differs from the corresponding exponent for the droplet initial condition recently rigorously established in Ferrari and Occelli (2018), Basu and Ganguly (2018), and requires novel arguments. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are expected to be applicable for a wider class of initial data.
For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, consider the geodesic $Γ_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of $Γ_n$ around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for $Γ_{n}$ and establish that for small $δ$, the probability that $Γ_{n}$ is contained in a strip of width $δn^{2/3}$ around the diagonal is $\exp (-Θ(δ^{-3/2}))$ uniformly in high $n$. We also obtain optimal small deviation estimates for the one point distribution of the geodesic showing that for $\frac{t}{2n}$ bounded away from $0$ and $1$, we have $\mathbb{P}(|x(t)-y(t)|\leq δn^{2/3})=Θ(δ)$ uniformly in high $n$, where $(x(t),y(t))$ is the unique point where $Γ_{n}$ intersects the line $x+y=t$. Our methods are expected to go through for other exactly solvable models of planar last passage percolation and, upon taking the $n\to \infty$ limit, provide analogous estimates for geodesics in the directed landscape.