arXiv subjects

Subhajit Goswami

Publications and source records attributed to Subhajit Goswami.

At least 19 recordsLinked to original sources

arXiv2026

Supercritical level-set percolation on finite graphs

We continue the study of bond percolation on finite graphs induced by the level-sets of the zero-average Gaussian free field on the associated metric graph. In our previous work Goswami and Pal (2026), we analyzed the critical and near-critical behavior of this model on uniform expanders, focusing on the volume of the largest cluster. In the present paper, we investigate the size of the second largest cluster in the supercritical regime --- the phase where one expects a unique giant component. We prove that the second largest cluster is at most of logarithmic order with high probability. The estimates hold uniformly over deterministic graphs with a bounded maximum degree and a spectral gap bounded away from zero. In particular, we do not require any assumption regarding regularity, girth or local geometry. This picture is substantially different from Bernoulli percolation, where the second largest supercritical cluster can have polynomial volume under identical hypotheses. The proof relies on expansion properties of the underlying graph and the Dirichlet martingale framework developed in Goswami and Pal (2026).

math.PR↗

arXiv2026

Sharp minimax risks and phase transitions in sparse submatrix detection

In this paper, we study the minimax risk of detecting a sparse, elevated-mean Gaussian submatrix planted in a larger noisy matrix. When the planted submatrix has size $n\times n$ and the ambient matrix has size $N\times N$ with $N=n^{1+α}$, the seminal work of Butucea and Ingster (2013) identifies the sharp detection boundary around which the risk converges to $0$ or $1$. We extend that zero-one theory by determining sharp risk asymptotics throughout a two-parameter phase diagram. Above the boundary, the risk decays stretched- or super-exponentially fast, and we determine its precise exponent. Below the boundary, where the risk tends to $1$, we obtain matching absolute-constant bounds for its polynomial rate of convergence. In both regimes, the asymptotics change across the line $α+δ=1/2$, where $δ$ parametrizes the signed distance from the detection boundary. Finally, on the boundary, we show that the minimax risk converges to the nondegenerate constant $\frac12$ in a very sparse regime where $n$ remains fixed and $N\to\infty$. Each of these rates corresponds to the risk of a suitably calibrated scan or sum test, whence follow the upper bounds. The main problem is to derive the matching regime-specific lower bounds. The mixture likelihood ratio is an average rather than a product and thus its logarithm lacks the additive structure required by standard large deviation formulations. On the other hand, classical second-moment estimates fail to be sharp due to overlap-induced correlations. To address these issues, we develop two general lower-bound principles based on truncated mixture likelihoods, tailored to the target risks on both sides of the detection boundary. The same framework yields sharp risk asymptotics for sparse subtensors with equal side lengths and also applies to more general testing problems including the detection of rectangular subtensors.

math.ST↗

arXiv2026

Sharp connectivity bounds for the vacant set of random interlacements

We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal exponential order for all $u$ in dimension three and all $u \neq u_\ast$ in higher dimensions, where $u_*$ refers to the critical parameter of the model, and they match the upper bounds derived in the article arXiv:2503.14497. In dimension three, our results further imply that the truncated two-point function grows at large distances $x$ at a rate that depends on $x$ only through its Euclidean norm, which offers a glimpse of the expected (Euclidean) invariance of the scaling limit at criticality. The rate function is atypical, it incurs a logarithmic correction and comes with an explicit pre-factor that converges to $0$ as the parameter $u$ approaches the critical point $u_*$ from either side. A particular challenge stems from the combined effects of lack of monotonicity due to the truncation in the super-critical phase, and the precise (rotationally invariant) controls we seek, that measure the effects of a certain "harmonic humpback" function. Among others, their derivation relies on rather fine estimates for hitting probabilities of the random walk in arbitrary direction $e$, which witness this invariance at the discrete level, and preclude straightforward applications of projection arguments.

math.PR↗

arXiv2026

Critical level-set percolation on finite graphs and spectral gap

We study the bond percolation on finite graphs induced by the level-sets of zero-average Gaussian free field on the associated metric graph above a given height (level) parameter $h \in \mathbb{R}$. We characterize the near- and off-critical phases of this model for any expanders family $\mathcal{G}_n = (V_n, E_n)$ with uniformly bounded degrees. In particular, we show that the volume of the largest open cluster at level $h_n$ is of the order $|V_n|^{\frac23}$ when $h_n$ lies in the corresponding critical window which we identify as $|h_n| = O(|V_n|^{-\frac13})$. Outside this window, the volume starts to deviate from $Θ(|V_n|^{\frac23})$ culminating into a linear order in the supercritical phase $h_n = h < 0$ (the giant component) and a logarithmic order in the subcritical phase $h_n = h > 0$. We deduce these from effective estimates on tail probabilities for the maximum volume of an open cluster at any level $h$ for a generic base graph $\mathcal{G}$. The estimates depend on $\mathcal{G}$ only through its size and upper and lower bounds on its degrees and spectral gap respectively. To the best of our knowledge, this is the first instance where a mean-field critical behavior is derived under such general setup for finite graphs. The generality of these estimates preclude any local approximation of $\mathcal{G}$ by regular infinite trees -- a standard approach in the area. Instead, our methods rely on exploiting the connection between spectral gap of the graph $\mathcal{G}$ and its connection to the level-sets of zero-average Gaussian free field mediated via a set function we call the zero-average capacity.

math.PR↗

arXiv2025

Strong local uniqueness for the vacant set of random interlacements

We consider the the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$ in dimensions $d \ge 3$. For varying intensity $u > 0$, the connectivity properties of $\mathcal V^u$ undergo a percolation phase transition across a critical parameter $u_* \in (0,\infty)$. In this article, we prove that this phase transition is sharp in the supercritical phase $u < u_*$. This follows from a certain strong local uniqueness property (SLU) introduced in the present work, which we prove $\mathcal{V}^u$ satisfies. In itself, this property furnishes the missing ingredient needed to deduce a number of desirable quenched results characterizing the large-scale geometry of the infinite cluster. Moreover, SLU entails a sought-after local and monotone criterion amenable to renormalization arguments below $u_*$.

math.PR↗

arXiv2025

A new approach to locally adaptive polynomial regression

Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for $\ell_0$-penalized regression. Although similar in spirit to Lepski's method and its variants in selecting the largest interval satisfying an admissibility criterion, our approach stems from a distinct philosophy, utilizing criteria based on $\ell_2$-norms of interval projections rather than explicit point and variance estimates. We obtain non-asymptotic risk bounds for the local polynomial regression methods based on our bandwidth selection procedure which adapt (near-)optimally to the local Hölder exponent of the underlying regression function simultaneously at all points in its domain. Furthermore, we show that there is a single ideal choice of a global tuning parameter in each case under which the above-mentioned local adaptivity holds. The optimal risks of our methods derive from the properties of solutions to a new ``bandwidth selection equation'' which is of independent interest. We believe that the principles underlying our approach provide a new perspective to the classical yet ever relevant problem of locally adaptive nonparametric regression.

stat.ML↗

arXiv2024

Equality of critical parameters for percolation of Gaussian free field level-sets

We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.

math.PR↗

arXiv2023

Phase transition for the vacant set of random walk and random interlacements

We consider the set of points visited by the random walk on the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$, for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left by the walk undergoes a phase transition across a non-degenerate critical value $u_* = u_*(d)$, as follows. For all $u< u_*$, the vacant set contains a giant connected component with high probability, which has a non-vanishing asymptotic density and satisfies a certain local uniqueness property. In stark contrast, for all $u> u_*$ the vacant set scatters into tiny connected components. Our results further imply that the threshold $u_*$ precisely equals the critical value, introduced by Sznitman in arXiv:0704.2560, which characterizes the percolation transition of the corresponding local limit, the vacant set of random interlacements on $\mathbb{Z}^d$. Our findings also yield the analogous infinite-volume result, i.e. the long purported equality of three critical parameters $\bar u$, $u_*$ and $u_{**}$ naturally associated to the vacant set of random interlacements.

math.PR↗

arXiv2023

A characterization of strong percolation via disconnection

We consider a percolation model, the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$, $d \geq 3$, in the regime of parameters $u>0$ in which it is strongly percolative. By definition, such values of $u$ pinpoint a robust subset of the super-critical phase, with strong quantitative controls on large local clusters. In the present work, we give a new charaterization of this regime in terms of a single property, monotone in $u$, involving a disconnection estimate for $\mathcal{V}^u$. A key aspect is to exhibit a gluing property for large local clusters from this information alone, and a major challenge in this undertaking is the fact that the conditional law of $\mathcal{V}^u$ exhibits degeneracies. As one of the main novelties of this work, the gluing technique we develop to merge large clusters accounts for such effects. In particular, our methods do not rely on the widely assumed finite-energy property, which the set $\mathcal{V}^u$ does not possess. The charaterization we derive plays a decisive role in the proof of a lasting conjecture regarding the coincidence of various critical parameters naturally associated to $\mathcal{V}^u$ in a companion article.

math.PR↗

arXiv2023

Finite range interlacements and couplings

In this article, we consider the interlacement set $\mathcal{I}^u$ at level $u>0$ on $\mathbb{Z}^d$, $d \geq3$, and its finite range version $\mathcal{I}^{u,L}$ for $L >0$, given by the union of the ranges of a Poisson cloud of random walks on $\mathbb{Z}^d$ having intensity $u/L$ and killed after $L$ steps. As $L\to \infty$, the random set $\mathcal{I}^{u,L}$ has a non-trivial (local) limit, which is precisely $\mathcal{I}^u$. A natural question is to understand how the sets $\mathcal{I}^{u,L}$ and $\mathcal{I}^{u}$ can be related, if at all, in such a way that their intersections with a box of large radius $R$ almost coincide. We address this question, which depends sensitively on $R$, by developing couplings allowing for a similar comparison to hold with very high probability for $\mathcal{I}^{u,L}$ and $\mathcal{I}^{{u'},2L}$, with $u' \approx u$. In particular, for the vacant set $\mathcal{V}^u=\mathbb{Z}^d \setminus \mathcal{I}^u$ with values of $u$ near the critical threshold, our couplings remain effective at scales $R \gg \sqrt{L}$, which corresponds to a natural barrier across which the walks of length $L$ comprised in $\mathcal{I}^{u,L}$ de-solidify inside $B_R$, i.e. lose their intrinsic long-range structure to become increasingly "dust-like". These mechanisms are complementary to the solidification effects recently exhibited in arXiv:1706.07229. By iterating the resulting couplings over dyadic scales $L$, the models $\mathcal{I}^{u,L}$ are seen to constitute a stationary finite range approximation of $\mathcal{I}^u$ at large spatial scales near the critical point $u_*$. Among others, these couplings are important ingredients for the characterization of the phase transition for percolation of the vacant sets of random walk and random interlacements in two upcoming companion articles.

math.PR↗

arXiv2023

Central Limit Theorem for Gram-Schmidt Random Walk Design

We prove a central limit theorem for the Horvitz-Thompson estimator based on the Gram-Schmidt Walk (GSW) design, recently developed in Harshaw et al.(2022). In particular, we consider the version of the GSW design which uses randomized pivot order, thereby answering an open question raised in the same article. We deduce this under minimal and global assumptions involving only the problem parameters such as the (sum) potential outcome vector and the covariate matrix. As an interesting consequence of our analysis we also obtain the precise limiting variance of the estimator in terms of these parameters which is smaller than the previously known upper bound. The main ingredients are a simplified skeletal process approximating the GSW design and concentration phenomena for random matrices obtained from random sampling using the Stein's method for exchangeable pairs.

math.ST↗

arXiv2022

On the radius of Gaussian free field excursion clusters

We consider the Gaussian free field $φ$ on $\mathbb{Z}^d$, for $d\geq3$, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set $\{φ\geq h\}$ exceeds a large value $N$, for any height $h \neq h_*$, where $h_*$ refers to the corresponding percolation critical parameter. In dimension $d=3$, we prove that this probability is sub-exponential in $N$ and decays as $\exp\{-\fracπ{6}(h-h_*)^2 \frac{N}{\log N} \}$ as $N \to \infty$ to principal exponential order. When $d\geq 4$, we prove that these tails decay exponentially in $N$. Our results extend to other quantities of interest, such as truncated two-point functions and the two-arms probability for annuli crossings at scale N.

math.PR↗

arXiv2022

Roughness of geodesics in Liouville quantum gravity

The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In the current article we confirm the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is stirctly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter $ξ$. We deduce this from a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). We also prove an analogous result for the LFPP geodesics.

math.PR↗

arXiv2022

Spatially Adaptive Online Prediction of Piecewise Regular Functions

We consider the problem of estimating piecewise regular functions in an online setting, i.e., the data arrive sequentially and at any round our task is to predict the value of the true function at the next revealed point using the available data from past predictions. We propose a suitably modified version of a recently developed online learning algorithm called the sleeping experts aggregation algorithm. We show that this estimator satisfies oracle risk bounds simultaneously for all local regions of the domain. As concrete instantiations of the expert aggregation algorithm proposed here, we study an online mean aggregation and an online linear regression aggregation algorithm where experts correspond to the set of dyadic subrectangles of the domain. The resulting algorithms are near linear time computable in the sample size. We specifically focus on the performance of these online algorithms in the context of estimating piecewise polynomial and bounded variation function classes in the fixed design setup. The simultaneous oracle risk bounds we obtain for these estimators in this context provide new and improved (in certain aspects) guarantees even in the batch setting and are not available for the state of the art batch learning estimators.

math.ST↗

arXiv2021

One-Shot Coordination of First and Last Mode Transportation

In this paper, we consider coordinated control of feeder vehicles for first and last mode transportation. The model is macroscopic with volumes of demands and supplies along with flows of vehicles. We propose a one-shot problem for transportation of demand to or from a hub within a fixed time window, assuming the knowledge of the demand and supply configurations. We present a unified optimization framework that is applicable for both operator profit maximization and social welfare maximization. The latter goal is useful for applications such as disaster response. The decision variables in the optimization problem are routing and allocations of the vehicles for different services. With K.K.T. analysis we propose an offline method for reducing the problem size. Further, we also analyze the problem of maximizing profits by optimally locating the supply for a given total supply and present a closed form expression of the maximum profits that can be earned over all supply configurations for a given demand configuration. We also show an equivalence between optimal supply location in the first mode problem and the last mode problem. We present a model for pricing based on the cost and travel time of the best alternate transportation and present necessary conditions for the feeder service to be viable. We illustrate the results through simulations and also compare the proposed model with a traditional vehicle routing problem. Through simulations, we also compare with the microscopic version of the problem with the decision variables being integers. We demonstrate that the route reduction algorithm proposed for the macroscopic formulation is still useful for computing nearly optimal solutions to the microscopic problem with much improved computational efficiency.

eess.SY↗

arXiv2021

New Risk Bounds for 2D Total Variation Denoising

2D Total Variation Denoising (TVD) is a widely used technique for image denoising. It is also an important nonparametric regression method for estimating functions with heterogenous smoothness. Recent results have shown the TVD estimator to be nearly minimax rate optimal for the class of functions with bounded variation. In this paper, we complement these worst case guarantees by investigating the adaptivity of the TVD estimator to functions which are piecewise constant on axis aligned rectangles. We rigorously show that, when the truth is piecewise constant, the ideally tuned TVD estimator performs better than in the worst case. We also study the issue of choosing the tuning parameter. In particular, we propose a fully data driven version of the TVD estimator which enjoys similar worst case risk guarantees as the ideally tuned TVD estimator.

math.ST↗

arXiv2021

Adaptive Estimation of Multivariate Piecewise Polynomials and Bounded Variation Functions by Optimal Decision Trees

Proposed by Donoho (1997), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal decision tree estimators fit piecewise polynomials of any given degree. Like Dyadic CART in two dimensions, we reason that these estimators can also be computed in polynomial time in the sample size $N$ via dynamic programming. We prove oracle inequalities for the finite sample risk of Dyadic CART and ORT which imply tight risk bounds for several function classes of interest. Firstly, they imply that the finite sample risk of ORT of order $r \geq 0$ is always bounded by $C k \frac{\log N}{N}$ whenever the regression function is piecewise polynomial of degree $r$ on some reasonably regular axis aligned rectangular partition of the domain with at most $k$ rectangles. Beyond the univariate case, such guarantees are scarcely available in the literature for computationally efficient estimators. Secondly, our oracle inequalities uncover minimax rate optimality and adaptivity of the Dyadic CART estimator for function spaces with bounded variation. We consider two function spaces of recent interest where multivariate total variation denoising and univariate trend filtering are the state of the art methods. We show that Dyadic CART enjoys certain advantages over these estimators while still maintaining all their known guarantees.

math.ST↗

arXiv2020

Existence of phase transition for percolation using the Gaussian Free Field

In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in the sense that $p_c<1$). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasi-transitive graphs (in particular, Cayley graphs) with super-linear growth is strictly smaller than 1, thus answering a conjecture of Benjamini and Schramm. The proof relies on a new technique consisting in expressing certain functionals of the Gaussian Free Field (GFF) in terms of connectivity probabilities for percolation model in a random environment. Then, we integrate out the randomness in the edge-parameters using a multi-scale decomposition of the GFF. We believe that a similar strategy could lead to proofs of the existence of a phase transition for various other models.

math.PR↗