Papers

We prove the analogue of Schramm's locality conjecture for long-range percolation on transitive graphs of polynomial growth with $\alpha \in (0,2)$. In this setting, we also prove the joint continuity of the percolation probability $\theta$ with respect to three parameters: the underlying graph with respect to the local topology, the connectivity kernel, and the percolation parameter $\beta$ for all values of $\beta \in \mathbf{R}_+$, including the critical parameter $\beta_c$. We also prove a number of results related to the supercritical sharpness of long-range percolation: the long-range order decay of the distribution of finite clusters, the truncation problem, the anchored isoperimetric dimension and the transience of the infinite percolation cluster, and the smoothness of the percolation characters. We obtain these results from proving the local existence-and-uniqueness of the linear-sized (giant) cluster. As an immediate corollary of the local existence-and-uniqueness of the giant we obtain the law of large numbers, which answers a special case of a question of Nekrashevych and Pete \cite[Question 1.3]{nekrashevych_scale-invariant_2011}. The main technical contribution is the construction of a renormalisation scheme combining iteratively merged Voronoi tiles with scale-invariant nets, related to the scale-invariant groups of Benjamini.

We investigate the asymptotic behavior and turnpike properties of graphon mean field control (GMFC) problems in the linear-quadratic setting. We consider both a finite-horizon GMFC problem and its associated ergodic counterpart, in which the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous interactions. The optimal controls and state trajectories for both problems are characterized by systems of Riccati equations together with systems of generalized differential and algebraic equations on suitable Hilbert spaces. Under a stabilizability condition and appropriate positivity assumptions on the graphon-induced operators, we establish the unique solvability of the ergodic control problem and derive exponential convergence estimates for the finite-horizon system to its stationary limit. As a consequence, we establish an exponential turnpike property for the optimal pair and prove the convergence of the time-averaged value function for the finite-horizon GMFC problem.

We study the emergence of (cross-scale) fluxes associated with energy and enstrophy in a stochastic version of Zeitlin's $SU(N)$ approximation of 2-d Euler dynamics. Motivated by the Euler-Arnold formulation, we interpret the nonlinear transport as motion along coadjoint orbits, reflecting an underlying symmetry that preserves all Casimir invariants. We introduce a fast-slow stochastic framework in which rapid mixing is modeled by a structured fast stochastic forcing acting along selected directions. We show that when the fast dynamics preserves the full coadjoint orbit symmetry given by the natural symplectic structure, the averaged system exhibits no nontrivial flux. In contrast, when this symmetry is broken by tangential non-Hamiltonian vector fields on the coadjoint orbit, we identify conditions that yield nonzero fluxes carried by the Euler-Arnold nonlinearity. Thus, coadjoint symmetry suppresses averaged nonlinear flux, whereas its breaking can select a preferred direction of cross-scale transfer.

We show that the Belopol'skaya-Daletskii formulation of stochastic differential equations on a Riemann manifold offers an elementary way to construct equivariant representations of finite-dimensional approximations to the path measure of a diffusion. The key ingredient is the use of the exponential map to describe increments of the diffusion.

We show a Liouville theorem for domains in graphs with Dirichlet boundary conditions. More specifically, we characterize the non-existence of non-zero bounded harmonic functions. Since Dirichlet boundary conditions give rise to Laplacians with a positive killing term, we can first characterize the validity of the Liouville theorem by the fact that the Green operator applied to the killing terms is equal to 1, or in other words, that the constant function $1$ is a potential. Secondly, we derive a characterization in terms of stochastic completeness at infinity and and total loss of heat. Thirdly, we give a characterization in terms of a Green formula for superharmonic potentials. Finally, we investigate the Liouville property in terms of recurrence and transience of the graph without killing term. As an application, we consider subsets of the Euclidean space such as cones and percolation clusters, and weakly spherically symmetric graphs.

We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\mathbb{H}^d$ for $d\geq 3$. By recent results of Grebík and Recke and d'Achille et al., this "uniqueness threshold" $p_u(\lambda)$ tends to zero as the intensity $\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}$ with $k,d_1,\dots,d_k\geq 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane $\mathbb{H}^2$, Benjamini and Schramm have shown that $p_u(\lambda)$ tends to one as $\lambda$ tends to zero, and $p_u(\lambda)>1/2$ for all $\lambda>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$, the uniqueness threshold satisfies $p_u(\lambda)\leq 1/2$ for all $\lambda>0$. Here we will show that $\inf_{\lambda>0}p_u(\lambda)>0$ for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$. This answers a question of Grebík and Recke.

We consider an insurance risk model with a random walk structure in which the underlying probability law is unknown. A decision maker observes a statistic $Q_n$ computed from $n$ historical observations and then needs to choose a capital buffer of the form $C_n=n f(c,Q_n)$, where $c=\log(1/\delta)/n$ balances the amount of data and the tolerated ruin probability $\delta$. The classical safe capital buffer is inversely proportional to the adjustment coefficient $\gamma$, the exponential rate at which the ruin probability decays; when the law is unknown, this coefficient must be inferred from the data. The profile $f$ couples the statistical cost of observing an atypical historical statistic with the future ruin exponent induced by the resulting decision. We study the joint ex ante probability (over both the historical sample and an independent future risk process) that the future maximum exceeds the data-dependent buffer. We show that the naive plug-in rule fails to achieve the prescribed logarithmic decay exponent $c$, illustrating the adverse impact of model uncertainty when making decisions under rare-event constraints. We then identify a profile $f^*=f^*(c,Q_n)$ with the following appealing properties: its lower semicontinuous majorants are safe, while regular continuous rules that fall strictly below $f^*$ are, under mild conditions, unsafe. We illustrate the potential applicability of our framework by developing three parametric examples. In nonparametric settings, we show that a single exponential envelope leads to degenerate capital buffers, whereas a two-level exponential envelope yields a nondegenerate capital buffer which we prove to be safe.

A strong law of large numbers (SLLN) for sublinear expectation under a certain integrability condition is proved. It is shown that the proposed integrability condition is weaker than the majorant condition introduced by Cheng Hu (2018). The proof is based on the truncation method of N. Etemadi (1981).

We study free infinite divisibility in the two-parameter family of beta distributions $\{\beta_{p,q}:p,q>0\}$. Conditional positive definiteness of free cumulants yields a hierarchy of necessary Hankel conditions. We factor the first nontrivial determinant and obtain the explicit necessary inequality \[ 2s^3(s+1)+pq\bigl(s^3-7s^2-16s-12\bigr)\geq0, \qquad s=p+q. \] Its strict reverse defines an open two-dimensional non-freely-infinitely-divisible region not contained in the previously known exclusions. As a boundary consequence, we complete the classification of one boundary family: $\beta_{1/2,q}$ is freely infinitely divisible if and only if $q\geq3/2$. The $3\times3$ determinant is also obtained explicitly in the symmetric variables $s=p+q$ and $u=pq/s^2$. Finally, exact-rational $LDL^{\mathsf T}$ certificates show that each leading Hankel test from $3\times3$ through $12\times12$ strictly enlarges the exclusion supplied by all preceding leading tests. In particular, the $4\times4$ test already detects an open set with $p+q>3$, beyond the range accessible to the $2\times2$ determinant. The results are finite-order obstructions rather than a complete classification; two limiting arguments explain why no fixed member $H_N$ of the leading Hankel hierarchy can provide a uniform obstruction up to the small-parameter boundary.

Given a random variable $X$, an independent random variable $Y$ is called a symmetrizer of $X$ if their sum $X+Y$ is symmetric about the origin. The study of symmetrization resistance asks whether every such $Y$ must contain at least as much randomness as $X$. This problem was previously investigated for binary random variables in terms of variance and Shannon entropy. In this paper, we establish sharp symmetrization resistance results for exponential and geometric distributions. We prove that every independent symmetrizer $Y$ of an exponential random variable $X$ has variance, Rényi and Tsallis entropies of every positive order at least as large as that of $X$. Parallel results are obtained for integer-valued independent symmetrizers of geometric random variables. Equality in each comparison holds precisely when $Y$ is an independent copy of $-X$. Furthermore, we show the majorization of $X$ over $Y$ via establishing sharp concentration inequalities for $Y$. The proofs crucially rely on a differential/difference inversion formula for exponential/geometric convolution, which converts the symmetrization constraint into a hazard-rate inequality and enables us to identity exponential/geometric distributions as extremal distributions of a corresponding optimization problem.

Motivated by confidence-guided parallel unmasking in masked discrete diffusion, we study a single selection step in a stylized Gaussian random-field model. A locally dependent nonnegative score field represents position wise uncertainty, and the scheduler selects the K positions with the smallest scores. Dependence among the selected positions is measured through a distance-dependent Gaussian correlation model. This separation provides a tractable framework for quantifying how the geometry of low-score locations affects the dependence cost of factorized parallel decoding. We establish two complementary results. In a conservative sub-square-root regime, the conditional Gaussian total correlation of the selected block vanishes in probability. At the square-root scale, it remains non-negligible with positive asymptotic probability and admits a strictly positive expectation lower bound. Synthetic experiments support the predicted finite-size behavior. These results provide a rigorous stochastic-geometry baseline for understanding how budget size, score dependence, and spatial correlation jointly shape one-step confidence-based selection in masked discrete diffusion.

How much does an algorithm's running-time distribution under independent randomness reveal about its behavior when independence is no longer guaranteed? We study sources satisfying $\nu[w]\le DP[w]^s$ for every finite prefix $w$, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$. The constraint controls complete-prefix probabilities while allowing individual choices to be predictable, even fully determined by the past. For retry tasks, all deterministic history-dependent selectors have the same independent-source running-time law. Yet two orders have worst-case failure probabilities $1$ and $\exp[-\Theta(n)]$ at the same linear deadline under the same source constraint. We identify a static priority rule that is optimal at every deadline and every $D$. For the standard local walk on a $k$-CNF with at least $r$ true literals per clause under some assignment, $k/2<r<k$, we determine the sharp source threshold $s_*$. At and above it, the expected flip count is $O_{k,r}(\min\{L^3,L/(s-s_*)\})$, where $L=h+\log D+1$, $h$ is the initial Hamming distance to that assignment, and $L/0=\infty$. The bound allows arbitrary clause overlap and history-dependent clause selection. Matching instances admit one source forcing this delay with probability one for every selector. At criticality and fixed $D$, the delay is cubic despite a linear independent-source expectation. Variable-depth prefix covers, together with classical tree max-flow/min-cut, yield an exact criterion for restoring exponential tails by restarting on the same tape. We synthesize updates and restarts for explicit finite-state processes. Under a sufficient prefix guarantee, we also obtain noisy predecessor search with error at most $\eta$ and expected query count polynomial in the correct leaf's depth and $\log(D/\eta)$, without knowing the depth or tree height.

We establish pathwise large-deviation bounds for the empirical-measure flow of a cluster coagulation model with gelling kernels on a locally compact Polish state space. The cluster coagulation process is a measure-valued Markov jump process in which pairs of clusters merge according to a time-inhomogeneous kernel $K(t,x,y,d z)$. Kernels with sufficiently rapid growth may produce gelation: a phase transition in which mass escapes from the finite-cluster population in finite time. We first prove tightness of the empirical flows and identify every subsequential limit as a solution of the time-inhomogeneous multi-type Flory equation, which accounts for post-gelation dynamics through a conserved quantity. We then establish a global large-deviation upper bound on path space with a variational rate function. We also prove a local lower bound near paths that are unique solutions of suitable tilted Flory equations. Finally, we give a convex-dual representation of the upper rate function, including the possible singular part of the representing measure. Our results apply to locally compact Polish state spaces with continuous mass functions, encompassing both the gelling and non-gelling regimes within a unified framework.

This paper aims to investigate a diffusion approximation of a sequence of critical branching processes with strictly stationary and ergodic immigration. Under fairly general assumptions on the dependence structure of the immigration sequence, we establish that the scaled and properly normalized sequence of branching processes with immigration converges in distribution to the diffusion process in the Skorokhod topology. Our result extends the functional limit theorem of Wei and Winnicki (1989) to the case where the immigration sequence is dependent.

For $d\in\{1,2\}$, let $H=-\frac{1}{2}\Delta + V +\xi$ be the random Schrödinger operator on $L^2(\mathbb{R}^d)$ where $\xi$ is a standard Gaussian white noise and $V$ is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schrödinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of $\mathrm{Tr}[e^{-sH}]$ and $\mathrm{Tr}[e^{-tH}]$ as $s,t\to0$ through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case $d=1$ and are the first of their kind for $d=2$. As an application of these new estimates, we prove a quantitative hyperuniformity-type property and decorrelation rate for the trace as $s,t\to0$.

For a two-component log-concave mixture model, we investigate the extent to which the weight, mean, and covariance of each component distribution can be accurately recovered when given sufficient samples of the mixture distributions. One fundamental obstruction is that the mixture distribution itself could sometimes be log-concave, and in this case, accurate recovery is impossible. In this paper, we identify two regimes, one where the mixture distribution itself could be log-concave and another one where the mixture distribution is never log-concave, and one can always separate the two component distributions using a quadratic classifier.

We study the long-time, small-noise behavior of a class of dissipative stochastic evolution equations in a separable Hilbert space, driven by a cylindrical Wiener process whose covariance degenerates to a limiting operator in the strong operator topology. A prototypical example is a stochastic reaction-diffusion equation on a bounded domain with spatially homogeneous but spectrally regularized noise that becomes spatially rough in the limit. We establish a large deviation principle for occupation measures as the time horizon becomes large, the noise intensity tends to zero, and the noise becomes increasingly rough. The result covers a broad class of dissipative equations in infinite dimensions, including those driven by asymptotically rough cylindrical noise whose covariance need not be trace class. Extending the finite-dimensional work of Budhiraja and Zoubouloglou, the infinite-dimensional setting introduces substantial new difficulties: the bound arguments require careful handling of the unbounded evolution and inverse covariance operators, and the increasing roughness of the noise must be balanced against its vanishing amplitude through uniform estimates. Proofs combine analytic semigroup techniques, fractional domain space estimates and a careful treatment of stochastic convolutions in weighted spaces. The rate function is given by a simple explicit formula, the average over the measure of the squared Cameron-Martin cost of canceling the deterministic drift at each point. The proof follows the weak convergence approach based on the Bouè-Dupuis variational formula and constructs near-optimal controls by alternating travel phases that steer the process between prescribed target states and hold phases that stabilize it near a target while shaping the occupation measure.

This work investigates the mean-field stochastic PDEs involving a class of pseudo-monotone kernels. We first study the well-posedness – in both the strong and weak sense – within the variational framework by introducing a notion of measure-dependent pseudo-monotone operators, which generalizes the classical framework due to Brézis. Furthermore, we establish the quantitative dimension-free propagation of chaos within a $p$-uniformly convex Banach space for general infinite-dimensional weakly interacting systems, obtaining convergence rates that are near-optimal in a suitable sense. Our results reveal a new insight: the convergence rate of the mean-field limit is intrinsically governed by the geometry of the underlying solution space, specifically its modulus of convexity. As applications, we study several finite- and infinite-dimensional interacting particle systems arising in machine learning and fluid mechanics, including stochastic Stein variational gradient descent, mean-field Allen-Cahn equations, and Lagrangian-averaged Burgers equations.

We construct geometric regime-switching diffusions, a class of Markov processes on locally compact stratified Riemannian state spaces. In contrast with classical regime-switching and stochastic hybrid diffusions, the regimes are not external labels, but intrinsic strata of a singular geometric state space. Changes of regime may therefore change dimension, rank, geometry or combinatorial type while the state space maintains its ambient topology. On each stratum the motion is a conservative Feller diffusion, while inter-stratum transitions are specified by state-dependent jump rates and landing kernels along a directed graph. We characterize the process through a martingale problem on a natural stratified core. Under a uniform bound on the total jump rate, we construct a conservative càdlàg strong Markov process by combining the stratumwise diffusions with a Poisson thinning mechanism. Uniqueness is proved using an auxiliary disjoint-union topology and a bounded perturbation argument. Standard Foster–Lyapunov conditions for the extended generator give positive Harris recurrence, uniqueness of the invariant probability measure and, under aperiodicity, \(V\)-uniform geometric ergodicity. The framework is applied to the cone of positive semidefinite covariance matrices, stratified by rank. The resulting process combines fixed-rank covariance diffusions with stochastic rank changes and is \(V\)-uniformly geometrically ergodic.

For a class of non-autonomous linear SPDEs, we establish the equivalence among asymptotic regularization, weak observability, and approximate null controllability for the associated deterministic control systems. This equivalence provides a deterministic control-theoretic characterization of stochastic smoothing and offers a systematic approach to studying SPDEs driven by spatially localized noise. We further establish a criterion for semilinear SPDEs based on weak observability of the linearized equations. Our approach combines methods from PDE control theory with Malliavin calculus. As applications, we consider the stochastic Oseen equation, non-autonomous uniformly parabolic equations, and the parabolic Sine–Gordon equation, all driven by finite-dimensional, spatially localized white-in-time noise.

The randomness, efficiency (power and variability), and desirable allocation proportions are important components for evaluating a response-adaptive design in clinical trials and conflicted demands in applications. The aim of this paper is to provide designs dealing with these dilemmas. We first give a general framework for efficient response-adaptive randomization procedures that attain the Cramér-Rao lower bounds of the allocation variances for any desired allocation proportions. The general framework is flexible for us to define new families of efficient designs with good properties for both two and multiple-treatment clinical trials. We also prove that, among all response-adaptive randomization procedures with the same limit allocation proportions, the selection biases and entropies as measures of the randomness of the designs have their optimal values. Basing on the theory on efficiency and randomness, we propose a new family of doubly adaptive biased coin designs for multi-treatment clinical trials that can target any allocation proportion and are asymptotically best in terms both the randomness and efficiency so that their randomness is asymptotic optimal and asymptotic allocation variance attains the Cramér-Rao lower bound. Theoretical properties, including the strong consistency, the asymptotic normality, and the functional central limit theorem for both the sample allocation proportions and the estimators of the distribution parameters, are developed by using the technique of Gaussian approximation and Gaussian comparing theorems.

We establish quantitative two-time spatial decorrelation for the Kardar–Parisi–Zhang fixed point with flat initial data. For every $s,t>0$,there exist constants $C,c>0$ such that \[ \big|{\rm Cov}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\big| \le C\exp\{-c|x|^3\},\qquad |x|\ge1. \] Unlike the fixed-time covariance, which is governed directly by the Airy$_1$ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy$_1$ process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, the centered spatial averages, normalized by $N^{1/2}$, converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation of the flat KPZ fixed point.

In this paper, we establish transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by multiplicative space-time white noise. In particular, the transportation-cost inequalities are established with respect to the $L^1$ metric, the $L^2$ metric and the uniform metric under different dissipativity strengths of the drift. We first obtain a global-in-time and spatially uniform moment estimate for stochastic convolutions driven by space-time white noise, which is of independent interest. We then establish Lipschitz continuity of the solutions to the associated stochastic controlled equations with respect to the controls with a time-independent Lipschitz constant. Applying this Lipschitz continuity estimate, we finally prove transportation-cost inequalities for the invariant measures of stochastic reaction-diffusion equations.

Let $X_1, X_2, \ldots, X_n$ be independent random vectors. For a directed graph $G=(V,E)$ with vertex set $V=\{1,2,\ldots,n\}$ and a collection of bivariate kernels $\{h_e:e\in E\}$, we consider \[ U=\sum_{e=(i,j)\in E} h_e(X_i,X_j). \] This framework generalizes incomplete U-statistics by allowing the random vectors to be non-identically distributed, the kernels to be asymmetric and edge-dependent, and the sampling structure to be specified by an arbitrary graph. We derive several concentration inequalities for $U-\mathbb{E}U$. The main proof strategy exploits edge-coloring results from graph theory and relates the tail behavior of $U$ to the chromatic index of $G$. This approach is elementary, transparent, and readily adaptable to broader settings, including U-statistics of order $m>2$ and statistics involving doubly indexed random vectors.

We study stationary measures of the log-gamma polymer on a finite diagonal strip and in a half-space. We establish the phase diagram for the stationary measure on the strip. To this end we develop a representation of {the Laplace transform of} this stationary measure by independent $\mathrm{Beta}_{II}$ random variables. We also present an analytic approach that extends Barraquand's contour integral representation of the Laplace transform. We prove that, as the strip width tends to infinity, the stationary measure of the log-gamma polymer on the strip converges to the stationary measure of the half-space log-gamma polymer. Finally, we derive a contour integral formula for the Laplace transform of the stationary measure of the half-space log-gamma polymer.

For the nearest-neighbour Edwards–Anderson Ising model on the discrete torus, we prove existence of the quenched thermodynamic limit and almost-sure self-averaging of the free-energy density. The only moment assumption in the main argument is $\E|J|<\infty$, and no symmetry or centering of the coupling law is needed. The proof avoids periodic subadditivity: the wrap-around bonds form a surface-order perturbation, while a tiling argument and the strong law of large numbers give the free-boundary limit. We also obtain the quantitative bounds $(O(L^{-1})$ for the disorder-averaged finite-volume correction. A precise common probability space is specified, since an almost-sure statement across volumes is otherwise not well defined.

Recently, stochastic pantograph models have gained an intensive attention and have been used in different fields such as finance, biology, control and stochastic neural networks. It is also more preferable to incorporate jumps during the study of stochastic differential equations. In this paper, stochastic pantograph model with Poisson random measure is studied. The compensated split-step theta technique is applied to the considered model. The numerical scheme exhibits a non divergent attitude and converges to the solution of our model under assumptions addressed later on. Furthermore, the almost sure exponential stability of the numerical scheme is investigated via utilizing the discrete semi-martingale convergence theorem. Finally, theoretical findings are manifested via some numerical examples.

Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form $P(z)=p(z^m)$, where $p$ is a deterministic polynomial of degree $n$ with real, non-negative roots, in the regime where $m$ and $n$ are large. If $m\gg \log(n)$ and the root distribution of $P$ converges to a compactly supported, radial probability measure $\mu_0$, these works show that for $0\le t<1$, the root distribution of the $\lfloor nmt\rfloor$-th derivative of $P$ converges to a compactly supported probability measure $\mu_t$ given by an explicit formula for its radial quantile function. We give a substantially simplified proof of this result and also extend the result from repeated differentiation to repeated applications of the differential operator $z^a(d/dz)^b$. We also compute the limiting root distribution in the case when $m$ is fixed and $n$ tends to infinity.

We consider a novel robust utility maximisation problem under bounded cumulative transition rate uncertainty within the class of non-Markovian marked point processes on a finite state-space. Utility is maximised over the class of admissible controls, while Nature chooses a worst-case biometric scenario from the class of admissible, path-dependent cumulative transition rates restricted by path-dependent upper and lower bounds. We prove a martingale optimality principle and a novel existence and uniqueness result for a non-standard worst-case backwards stochastic differential equation, which allows us to establish existence and uniqueness of worst-case and best-case prospective reserves of life and health insurance contracts with reserve-dependent payments. Finally, we find an explicit solution of a novel robust consumption-insurance problem with power utility preferences.

We consider multipartite random graphs with given degree sequences, within and across different partitions. Under general assumptions, we prove the local limit of this graph is a multi-type branching process, establish that a giant component exists only when the local limit survives, and deduce that the typical distance is of logarithmic order in probability in the supercritical regime. Our analysis removes two major assumptions from Gamarnik and Misra (2015), where the giant component problem for this model was first considered. In particular, we do not assume irreducibility of the local limit, and provide a general framework to extract giant components even when the limiting branching process is reducible, which we hope to be useful in other contexts. We also provide a new simpler survival criterion of multi-type branching processes, which we hope to be useful when direct calculation of the spectral radius of the offspring matrix may prove to be difficult.