arXiv subjects

Dipranjan Pal

Publications and source records attributed to Dipranjan Pal.

5 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

The two-dimensional $\ell_{\infty}$ Directed Spanning Forest

We investigate the $\ell_{\infty}$ \textit{ directed spanning forest} (DSF), a directed forest whose vertex set is given by a homogeneous Poisson point process, in which each Poisson point connects to the nearest Poisson point, measured in $\ell_{\infty}$ distance, with a strictly larger $y$- coordinate. In this paper, we prove that the two-dimensional $\ell_{\infty}$ DSF is connected and obtain an optimal estimate on the tail decay of coalescing time of two $\ell_{\infty}$ DSF paths. Similar estimates were earlier obtained in \cite{MR4203342} for the $\ell_2$ (Euclidean) DSF and in \cite{garcia2025directed} for the $\ell_p$ DSF with $p \in [1, \infty]$ in two-dimension. The geometry of $\ell_{\infty}$ balls compels us to develop new 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

Edge spectra of Gaussian random symmetric matrices with correlated entries

We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$, with zero-mean, unit variance entries satisfying the condition $\sup_{(i, j) \ne (i', j')}|\mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + \varepsilon)})$, where $\varepsilon > 0$. It follows from Catalano et al. (2024) that the empirical spectral distribution of $n^{-1/2} X_n$ converges weakly almost surely to the standard semi-circle law. Using a Füredi-Komlós-type high moment analysis, we show that the largest eigenvalue $λ_1(n^{-1/2} X_n)$ of $n^{-1/2} X_n$ converges almost surely to $2$. This result is essentially optimal in the sense that one cannot take $\varepsilon = 0$ and still obtain an almost sure limit of $2$. We also derive Gaussian fluctuation results for the largest eigenvalue in the case where the entries have a common non-zero mean. Let $Y_n = X_n + \fracλ{\sqrt{n}}\mathbf{1} \mathbf{1}^\top$. When $\varepsilon \ge 1$ and $λ\gg n^{1/4}$, we show that \[ n^{1/2}\bigg(λ_1(n^{-1/2} Y_n) - λ- \frac{1}λ\bigg) \xrightarrow{d} \sqrt{2} Z, \] where $Z$ is a standard Gaussian. On the other hand, when $0 < \varepsilon < 1$, we have $\mathrm{Var}(\frac{1}{n}\sum_{i, j}X_{ij}) = O(n^{1 - \varepsilon})$. Assuming that $\mathrm{Var}(\frac{1}{n}\sum_{i, j} X_{ij}) = σ^2 n^{1 - \varepsilon} (1 + o(1))$, if $λ\gg n^{\varepsilon/4}$, then we have \[ n^{\varepsilon/2}\bigg(λ_1(n^{-1/2} Y_n) - λ- \frac{1}λ\bigg) \xrightarrow{d} σZ. \] While the ranges of $λ$ in these fluctuation results are certainly not optimal, a striking aspect is that different scalings are required in the two regimes $0 < \varepsilon < 1$ and $\varepsilon \ge 1$.

math.PR↗

arXiv2024

Spectra of adjacency and Laplacian matrices of Erdős-Rényi hypergraphs

We study adjacency and Laplacian matrices of Erdős-Rényi $r$-uniform hypergraphs on $n$ vertices with hyperedge inclusion probability $p$, in the setting where $r$ can vary with $n$ such that $r / n \to c \in [0, 1)$. Adjacency matrices of hypergraphs are contractions of adjacency tensors and their entries exhibit long range correlations. We show that under the Erdős-Rényi model, the expected empirical spectral distribution of an appropriately normalised hypergraph adjacency matrix converges weakly to the semi-circle law with variance $(1 - c)^2$ as long as $\frac{d_{\avg}}{r^7} \to \infty$, where $d_{\avg} = \binom{n-1}{r-1} p$. In contrast with the Erdős-Rényi random graph ($r = 2$), two eigenvalues stick out of the bulk of the spectrum. When $r$ is fixed and $d_{\avg} \gg n^{r - 2} \log^4 n$, we uncover an interesting Baik-Ben Arous-Péché (BBP) phase transition at the value $r = 3$. For $r \in \{2, 3\}$, an appropriately scaled largest (resp. smallest) eigenvalue converges in probability to $2$ (resp. $-2$), the right (resp. left) end point of the support of the standard semi-circle law, and when $r \ge 4$, it converges to $\sqrt{r - 2} + \frac{1}{\sqrt{r - 2}}$ (resp. $-\sqrt{r - 2} - \frac{1}{\sqrt{r - 2}}$). Further, in a Gaussian version of the model we show that an appropriately scaled largest (resp. smallest) eigenvalue converges in distribution to $\frac{c}{2} ζ+ \big[\frac{c^2}{4}ζ^2 + c(1 - c)\big]^{1/2}$ (resp. $\frac{c}{2} ζ- \big[\frac{c^2}{4}ζ^2 + c(1 - c)\big]^{1/2}$), where $ζ$ is a standard Gaussian. We also establish analogous results for the bulk and edge eigenvalues of the associated Laplacian matrices.

math.PR↗