arXiv subjects

Manan Bhatia

Publications and source records attributed to Manan Bhatia.

15 recordsLinked to original sources

arXiv2026

Uniformity of extremal behaviour along geodesics in Liouville quantum gravity

In random geometry, geodesics often have a tendency to coalesce together and traverse regions highly singular relative to typical environments. In this paper, working with the model of Liouville quantum gravity, we develop a technique which yields zero-one laws for many extremal statistics measured along interior segments of the geodesic. Namely, working with statistics such as the Euclidean dimension, optimal H\"older continuity exponents with respect to the Euclidean metric and the minimal/maximal encountered thickness for the underlying GFF, we obtain zero-one laws for the above for segments in the bulk of geodesics, thereby upgrading the results of Gwynne-Pfeffer-Sheffield '22. The primary technique used, developed in Bhatia-Kavvadias '25, is to lay down a large family of small-scale "typical" and well-behaved geodesics along interior segments of a long geodesic.

math.PR↗

arXiv2026

Dimension lower bounds in random geometry via Lipschitz functions

We prove lower bounds for the Hausdorff dimensions of various natural sets associated with the Liouville quantum gravity (LQG) metric. We prove that the set of 3-star points (i.e., starting points of three disjoint geodesics) has Hausdorff dimension at least two with respect to the LQG metric, which is conjectured to be optimal. Our proof works for a general class of planar length metrics which also includes, e.g., Kendall's Poisson roads metric. We additionally prove a dimension lower bound of one for the set of 2-star points intersected with the boundary and for the metric net intersected with the boundary, as well as a dimension lower bound of two for the intersection of two metric nets. In the particular setting of LQG, we obtain sharper lower bounds for the Hausdorff dimensions of the set of 2-star points and the LQG metric net, with respect to both the Euclidean metric and the LQG metric. Our proofs are primarily topological. The key idea is to express the sets of interest in terms of non-constancy sets of Lipschitz functions.

math.PR↗

arXiv2025

Strong confluence of geodesics in Liouville quantum gravity

$\gamma$-Liouville quantum gravity ($\gamma$-LQG) constitutes a family of planar random geometries whose geodesics exhibit intricate fractal behaviour. As is observed in various planar models of random geometry as part of the phenomenon of geodesic confluence, geodesics in $\gamma$-LQG tend to merge with each other. In particular, in Gwynne-Miller '19, it was established that in $\gamma$-LQG, geodesics targeted to a fixed point do coalesce in the sense that any two such geodesics almost surely merge before reaching their common target. However, in view of the randomness inherent to the geometry, it is a priori possible that while geodesics targeted to a fixed point do coalesce, there exists a sequence of geodesics $P_n$ converging to an exceptional geodesic $P$ as $n\rightarrow \infty$ such that $P_n$ does not overlap with $P$ for any $n$. In this paper, we prove that this is not possible, thereby establishing a strong confluence statement for $\gamma$-LQG for all $\gamma\in (0,2)$. This extends the results obtained in Miller-Qian '20 for $\gamma=\sqrt{8/3}$ to all subcritical values of $\gamma$. We discuss applications to the study of geodesic stars and geodesic networks and include a list of open questions.

math.PR↗

arXiv2025

Geodesic switches and exceptional times in dynamical Brownian last passage percolation

We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using $\Gamma_{(0,0)}^{(n,n),r}$ to denote a geodesic from $(0,0)$ to $(n,n)$ at time $r$, we prove that the expected total number of coarse-grained changes (or "switches") accumulated by $\Gamma_{(0,0)}^{(n,n),r}$ away from its endpoints during a time interval $[s,t]$ is at most $n^{5/3+o(1)}(t-s)$; we expect the exponent $5/3$ to be tight. Using the above estimate, we establish that the set $\mathscr{T}$ of exceptional times at which a non-trivial bi-infinite geodesic exists a.s. has Hausdorff dimension at most $1/2$. Further, for any fixed direction $\theta$, we show that the set $\mathscr{T}^\theta\subseteq \mathscr{T}$ of times at which a non-trivial bi-infinite geodesic directed along $\theta$ exists a.s. has Hausdorff dimension equal to $0$.

math.PR↗

arXiv2025

Near-existence of bigeodesics in dynamical exponential last passage percolation

It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set $\mathscr{T}$ of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that $\mathscr{T}$ is "very close" to being non-trivial; namely, we obtain an $\Omega( 1/\log n)$ lower bound on the probability that there exists a random time $t\in [0,1]$ at which a non-trivial geodesic of length $n$ passes through the origin at its midpoint; note that if the above probability were $\Omega(1)$, then it would imply the non-triviality of $\mathscr{T}$. We conjecture that, even if $\mathscr{T}\neq \emptyset$, it a.s. has Hausdorff dimension exactly zero.

math.PR↗

arXiv2024

Area measures and branched polymers in supercritical Liouville quantum gravity

We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge $Q \in (0,2)$ and central charge $\mathbf{c}_{\mathrm{L}} = 1+6Q^2 \in (1,25)$. Recent works have shown how to define LQG in this phase as a planar random geometry associated with a variant of the Gaussian free field, which exhibits "infinite spikes." In contrast, a number of results from physics, dating back to the 1980s, suggest that supercritical LQG surfaces should behave like "branched polymers": i.e., they should look like the continuum random tree. We prove a result which reconciles these two descriptions of supercritical LQG. More precisely, we show that for a family of random planar maps with boundary in the universality class of supercritical LQG, if we condition on the (small probability) event that the planar map is finite, then the scaling limit is the continuum random tree. We also show that there does not exist any locally finite measure associated with supercritical LQG which is locally determined by the field and satisfies the LQG coordinate change formula. Our proofs are based on a branching process description of supercritical LQG which comes from its coupling with CLE$_4$ (Ang and Gwynne, arXiv:2308.11832).

math.PR↗

arXiv2024

The metric removability of interfaces in the directed landscape

The directed landscape is a prominent model of random geometry which is believed to be the universal scaling limit of all planar random geometries in the Kardar-Parisi-Zhang universality class. It comes equipped with a few different natural simple curves associated to it, such as geodesics and interfaces. Given such a curve, one might wonder whether the geometry off this curve determines the entire landscape, or if in fact, there is non-trivial extra information actually present "on" the curve. In this paper, we show that the former is true for an interface in the directed landscape, while the latter is true for a geodesic instead. Further, as is used in the proof of the first assertion above, we show that the set of times where any geodesic intersects an interface a.s. has dimension zero.

math.PR↗

arXiv2023

The $d_γ/2$-variation of distance profiles in $γ$-Liouville quantum gravity

For Brownian surfaces with boundary and an interior marked point, a natural observable to consider is the distance profile, defined as the process of distances from the marked point to a variable point $x$ lying on the boundary. When the boundary is parametrized by the natural length measure on it, this distance profile turns out to be locally absolutely continuous to Brownian motion, and as a result, the boundary length measure itself has a natural interpretation as the quadratic variation process of the distance profile. In this paper, we extend this interpretation to $γ$-Liouville quantum gravity ($γ$-LQG), a one-parameter family of models of random geometry which is known to specialize to the case of Brownian geometry for the case $γ=\sqrt{8/3}$. With $d_γ$ denoting the Hausdorff dimension of $γ$-LQG, we show that for a $γ$-LQG surface with boundary, the natural boundary length measure can be interpreted (up to a constant factor) as the $d_γ/2$-variation process of the distance profile from an interior point.

math.PR↗

arXiv2023

A Peano curve from mated geodesic trees in the directed landscape

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.

math.PR↗

arXiv2023

When will (game) wars end?

We study several variants of the classical card game war. As anyone who played this game knows, the game can take some time to terminate, but it usually does. Here, we analyze a number of asymptotic variants of the game, where the number of cards is $n$, and show that all have expected termination time of order $n^2$. This is the same expected termination time as in the game where at each turn a fair coin toss decides which player wins a card, known as Gambler's Ruin and studied by Pascal, Fermat and others in the seventeenth century.

math.CO↗

arXiv2023

Duality in the directed landscape and its applications to fractal geometry

Geodesic coalescence, or the tendency of geodesics to merge together, is a hallmark phenomenon observed in a variety of planar random geometries involving a random distortion of the Euclidean metric. As a result of this, the union of interiors of all geodesics going to a fixed point tends to form a tree-like structure which is supported on a vanishing fraction of the space. Such geodesic trees exhibit intricate fractal behaviour; for instance, while almost every point in the space has only one geodesic going to the fixed point, there exist atypical points which admit two such geodesics. In this paper, we consider the directed landscape, the recently constructed scaling limit of exponential last passage percolation (LPP), with the aim of developing tools to analyse the fractal aspects of the tree of semi-infinite geodesics in a given direction. We use the duality (Pimentel '16) between the geodesic tree and the interleaving competition interfaces in exponential LPP to obtain a duality between the geodesic tree and the corresponding dual tree in the landscape. Using this, we show that problems concerning the fractal behaviour of sets of atypical points for the geodesic tree can be transformed into corresponding problems for the dual tree, which might turn out to be easier. In particular, we use this method to show that the set of points admitting two semi-infinite geodesics in a fixed direction a.s. has Hausdorff dimension $4/3$, thereby answering a question posed in Busani-Sepp\"{a}l\"{a}inen-Sorensen '22. We also show that the set of points admitting three semi-infinite geodesics in a fixed direction is a.s. countable.

math.PR↗

arXiv2022

Atypical stars on a directed landscape geodesic

In random geometry, a recurring theme is that any two geodesics emanating from a typical point part ways at a strictly positive distance from the above point, and we call such points as $1$-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this paper, we consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Vir\'ag '18 -- and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. We show that the above dimension is almost surely equal to $1/3$. This is in contrast to Ganguly-Zhang '22, where it was shown that set of atypical stars on the line $\{x=0\}$ has dimension $2/3$. This reduction of the dimension from $2/3$ to $1/3$ yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.

math.PR↗

arXiv2021

Environment seen from infinite geodesics in Liouville Quantum Gravity

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.

math.PR↗

arXiv2021

Small deviation estimates and small ball probabilities for geodesics in last passage percolation

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.

math.PR↗

arXiv2020

Moderate deviation and exit time estimates for stationary Last Passage Percolation

We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $ρ\in (0,1)$ and points $v_N=((1-ρ)^2 N,ρ^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $ρ=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on moderate deviation estimates for point-to-point and point-to-line exponential Last Passage Percolation which are obtained via random matrix estimates.

math.PR↗