Papers

In recent years, nonlinear dynamics derived from kinetic theory have gained attention in the context of sampling configurations of spin systems such as the Ising model. We focus on nonlinear dynamics for the hard-core model, a canonical spin system with hard constraints that specifies a distribution over independent sets in a graph, weighted by their sizes. We explore two distinct types of nonlinear dynamics: the mean-field dynamics, which preserves the density (or average size) of independent sets, and the single-site dynamics, which preserves the marginal vector (i.e., the occupancy probabilities of the vertices). These dynamics are natural stochastic processes for sampling from the hard-core model with a specified density or marginal vector, respectively, both of which are canonical instances of maximum entropy distributions that have been studied in various contexts. In contrast to linear Markov chains, there is a significant lack of a fundamental theoretical framework for nonlinear dynamics. We develop foundational theoretical tools for analyzing nonlinear dynamics within the context of the hard-core model. We establish almost linear convergence of both the mean-field and single-site dynamics at sufficiently low density through novel coupling arguments. We also establish exponential decay of relative entropy for the mean-field dynamics all the way up to the critical density. Additionally, we design new algorithms for sampling from the hard-core distribution with either a specified density or a specified marginal vector. These algorithms are based on a related linear Markov chain, called the particle-system dynamics and inspired by the so-called Kac's program, that approximates the associated nonlinear dynamics. As we demonstrate in the paper, they are comparable in time complexity, but simpler to implement, than traditional approaches based on learning parameter values.

Given a permutation $\sigma$, its corresponding inversion graph is obtained by adding an edge between $i<j$ if and only if $\sigma(i)>\sigma(j)$. The first results on random inversion graphs come from Acan and Pittel, who studied the connected threshold for a uniform permutation with fixed inversion number, and Bhattacharya and Mukherjee, who mostly focused on the degrees of the graph when the permutation is chosen uniformly at random. In this work, we call minimal inversion the minimal degree of the inversion graph and extend a theorem from Bhattacharya and Mukherjee to the case where the permutation is not only uniform, but obtained as the ordering of points sampled according to some distribution on the plane. Under regularity assumptions on the distribution, and for the appropriate $\alpha>0$, we show that the probability that the minimal inversion rescaled by $n^{\alpha/(\alpha+1)}$ is larger than $t$ behaves like $\exp(-ct^{\alpha+1})$ for some constant $c>0$ depending on the distribution. We further show that every $\alpha>0$ admits at least one corresponding distribution, thus proving that the minimal inversion can asymptotically scale as $n^\beta$ for any $\beta\in[0,1]$ (the cases $\beta=0$ and $\beta=1$ being obtained via the identity and anti-identity permutations, among others).

We study first-passage percolation on the $\ell$-spread-out line graph, where each vertex $i\in\{0,1,\dots,n\}$ is connected to all others at distance at most $\ell$. Here, we focus on the microscopic regime, with $\ell$ fixed as $n\to\infty$. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time $T_n$ from $0$ to $n$. If the weight distribution has finite variance or a heavy tail with exponent above $2/\ell_c$ where $\ell_c=\ell(\ell+1)/2$, then $T_n$ satisfies a Gaussian CLT with $\sqrt{n}$ scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that $T_n$, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of $T_n$.

The Watts-Strogatz random graph model on $n$ vertices with parameters $K$ (a positive even integer) and $p \in [0, 1]$ is constructed in two steps. First, one starts with a ring lattice on $n$ vertices, where each vertex is connected to its $K/2$ nearest neighbors on each side. Each edge in turn is then independently rewired with probability $p$ by replacing one endpoint with a uniformly chosen vertex not already adjacent to it. We study the empirical eigenvalue distribution of the adjacency matrix for this model, whose entries are highly dependent due to the rewiring construction. In the regime where both $K$ and $pK$ grow to infinity with the vertex size $n$, we show that, after appropriate scaling, the empirical eigenvalue distribution converges to the semicircle law. The proof is based on a novel coupling argument that approximates the adjacency matrix by a sum of two independent random matrices, one a sparse Wigner matrix and the other a random band matrix. In the case where $K$ and $p$ remain fixed, we propose conjectural formulas for the first five moments of the limiting eigenvalue distribution. These conjectures are supported by a convergence result relating the Watts-Strogatz model to another random graph model, together with numerical simulations.

For any finite family of Poisson multiple integrals and any finite $p\geq2$, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in $L^p$, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if $F$ lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between $F$ and a standard normal is bounded by $15.6(\mathbb{E}[F^4]-3)^{1/2}$. This removes Assumptions $\mathbf A$ and $\mathbf A^{\textbf{loc}}$ from the Kolmogorov bound of Döbler and Peccati (Ann. Probab., 2018). We also obtain quantitative $L^4$ estimates for all iterated Malliavin derivatives and, for $F$ in a Poisson chaos, the fourth moment assumption of $F$ forces the $L^4$-integrability of its kernel.

We investigate the long-time limit of an elephant random walk conditioned to avoid zero and show that the resulting process admits a Skorokhod-type embedding into a three-dimensional Bessel process. Using this embedding, we further establish several limit theorems for the conditioned elephant random walk, including a scaling limit to a deterministically rescaled and time-changed three-dimensional Bessel process, and a law of the iterated logarithm.

We analyze Reflected UAS routing for heterogeneous multi-server queues at fixed parameters under subcritical load. The deterministic surrogate is a reflected ODE on the nonnegative orthant, not the unconstrained drift equation. This reflected ODE has a unique boundary equilibrium characterized by a scalar consistency equation and a convex-potential representation; all trajectories converge to it. The older argument lifting deterministic Lyapunov descent to CTMC stability fails: the exact generator applied to the deterministic potential produces a boundary term absent from the reflected-ODE descent identity. We give a direct Foster-Lyapunov drift inequality for the CTMC using a weighted-quadratic function, bypassing the failed lift. At the benchmark parameter point, the boundary equilibrium matches the numerical attractor to machine precision, and the default Reflected UAS policy has lower mean queue length than UAS and JSSQ across independent seed blocks.

One of the outstanding questions in the theory of elephant random walks as observed by Gut and Stadtmüller (2023), is to determine how much memory is needed for a phase transition between the diffusive, critical and superdiffusive regimes to persist. To investigate this memory breakpoint, we introduce the tampered memory elephant random walk, in which the memory is partitioned into two disjoint sets $D_n$ and $D_n^c$, which may be deterministic or random. On $D_n^c$ the dynamics is the same as an elephant random walk, while on $D_n$ the increments are replaced by independent innovations. The resulting walk is thus driven by two competing components: elephant random walk and an independent simple random walk corresponding to the innovations. We first establish a law of large numbers when the increasing collections $\{D_n\}_{n \ge 1}$ and $\{D^c_n\}_{n \ge 1}$ have a renewal structure with exponential moments. We then identify a sharp threshold that governs the persistence of the phase transition for deterministic memory partitions. We show that if $\{D_n\}_{n \ge 1}$ is non-random increasing collection with increasing complement $\{D^c_n\}_{n \ge 1}$ such that $\lim_{n \to \infty} \frac{\lvert D^c_n\rvert}{n} >1/2$, then a phase transition into diffusive, critical and superdiffusive regimes persists, whereas for $\lim_{n \to \infty} \frac{\lvert D^c_n \rvert}{n}<1/2$, there is only the diffusive regime with $\mathcal{O}(\sqrt{n})$. The case of $\lim_{n\to \infty}\frac{\lvert D^c_n \rvert}{n}=1/2$ is also characterised. Thus, one-half emerges as the sharp breakpoint for the persistence of anomalous diffusion in this competitive setting. We conjecture that the same threshold governs the case when $\{D_n\}_{n \ge 1}$ is random. Our proofs rely on stochastic approximation applied to the two dependent competing components of the walk, representing the retained memory and the innovations.

In various disparate settings one studies how random processes give way to parabolic partial differential equations and in turn to functional inequalities involving gradients and variational calculus. Particles subject to gradient forces, the fields which move them, the empirical measures which they form, the smooth laws which those measures approximate, and differentiable functionals of these laws constitute a hierarchy of descriptions of diffusion. Intervening on this hierarchy is a choice of how a conintuity operator converts particle velocity to the evolution of measures. Here a central object is constructed mediating two different such descriptions. The space of smooth positive probability densities on a closed Riemannian manifold is treated as a Fréchet manifold whose full continuous cotangent space consists of nonconstant distributions and whose regular cotangent space consists of sufficiently regular nonconstant functions. Beginning from the operator on this space arising as the infinitesimal lift of diffeomorphisms of the base manifold, two different Hilbert completions of this space of densities account for a wide class of objects relevant, leading to a partial universalisation of the microscopic–mesoscopic–macroscopic hierarchy in Markov analysis.

In this note, using a result of Bernardi, Holden and Sun, we give an explicit geometric relation between the perimeter of the peeling process along the percolation interface of a triangulation, and the corresponding Kreweras walk. The relation naturally extends to the branching peeling exploration. This sheds light on the relation between the growth-fragmentation process and correlated Brownian excursions discovered by Da Silva, Powell and Watson.

We prove the redundancy of a family (the so-called melonic family) of large-$N$ moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator $C$ at large-$N$ has an expectation value that depends on $C$ exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.

Let $\mu(d x)\propto e^{-U(x)} d x$ on $\R^d$, where $U$ is $m$-strongly convex and $L$-smooth, and denote by $\kappa=L/m$ the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error $\varepsilon$, the expected query counts are $O(\kappa^{1/2}d\,(d\log\kappa+\log\frac1\varepsilon))$ gradient queries for the bouncy particle sampler and $O(\kappa d^{1/4}(d\log\kappa+\log\frac1\varepsilon))$ full-gradient equivalents for Zigzag, where $d$ coordinate-partial queries count as one equivalent.

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We establish the existence of kinetic solutions for continuous obstacles and further derive $L^1$-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin–Wentzell large deviation principle in $L^1(0,T;L^1(\mathbb{T}^N))$. Unlike existing large-deviation arguments based on control-uniform penalization of the obstacle, we work directly with the reflected skeleton equation and retain the possibly singular Radon reflection measure throughout, while a viscous approximation provides the spatial $H^1$-regularity required for compactness before passing to the inviscid limit. Our results provide a hyperbolic counterpart to the large deviation theory for parabolic reflected SPDEs developed by Matoussi, Sabbagh, and Zhang (2021).

We study parametric classes of almost stochastic dominance on general Polish spaces as order relations for probability distributions with a parameter $\gamma \in [0,1]$. Larger values of $\gamma$ correspond to weaker order relations: $\gamma=0$ gives classical stochastic dominance $\le_{st}$, whereas $\gamma=1$ gives a complete preorder based on comparison of expectations of a fixed increasing function $g$. It is well known that $X \le_{st} Y$ can be characterized by the existence of a solution to an optimal transport problem with $\mathrm{OT}_c(X,Y)=0$ for a suitable cost function $c$. We generalize this idea so that the best possible parameter $\gamma$ for almost stochastic dominance can be determined from the solution of an optimal transport problem. Using a generalization of the classical Kantorovich–Rubinstein duality theorem to quasi-pseudo-metrics, we derive a dual characterization of the order in terms of expectation comparisons for a parametric class of test functions. Consequently, our relations are always transitive, in contrast to some other recent approaches to almost stochastic dominance based on optimal transport. A natural multivariate approach to almost stochastic dominance, based on classes of test functions with bounds on partial derivatives, was recently introduced by Müller et al. (2025). We show that this approach is a special case of our framework and derive the best possible parameters $\gamma$ for examples considered there, as well as for other examples from the literature. We also prove a robustness result showing that, under small perturbations of the distributions in a Wasserstein-type metric related to the optimal transport problem, the best possible $\gamma$ increases only slightly.

This paper investigates the nonlinear stochastic fractional heat equation driven by a Gaussian noise that is white in time and fractional in space with a Hurst parameter $H \in \big(\frac{3-\alpha}{4}, \frac{1}{2}\big)$. Specifically, the driving operator is the fractional Laplacian of order $\alpha/2 \in (1/2, 1)$. We characterize the asymptotic behavior of the temporal increment $u(t+\varepsilon,x)-u(t,x)$ for fixed $t\ge 0$ and $x\in\mathbb{R}$ as $\varepsilon\downarrow 0$. Utilizing these precise asymptotic estimates, we establish Khinchin's and Chung's laws of the iterated logarithm for the temporal process $t \mapsto u(t,x)$.

We establish strong existence and pathwise uniqueness for McKean-Vlasov stochastic differential equations with coefficients satisfying a distribution-dependent Lyapunov condition. Under a hybrid Perron-Nagumo condition that permits a non-integrable singularity at the initial time, pathwise uniqueness holds within the class of strong solutions satisfying the corresponding Lyapunov estimate. For existence, we truncate the coefficients on nested bounded domains, construct absorbed local weak solutions, pass to a weak solution via tightness arguments, and then apply a restricted Yamada-Watanabe theorem to obtain a strong solution. Our existence proof, different from the classical truncation-patching method, is interesting in its own right. We also provide an explicit example to which our criterion applies, while none of the Lipschitz, Osgood, or monotonicity conditions is satisfied.

Let $F$ denote the Fano plane, the $3$-uniform hypergraph with $7$ vertices and $7$ edges. Frankl and Füredi, and independently Keevash and Sudakov, proved that the largest $F$-free subhypergraph of $K_n^{(3)}$ is bipartite. In this paper, we determine the sharp threshold for this property in the random setting. We show that for $\hat{p} = \Theta_F \cdot n^{-2/3} \left(\log n\right)^{1/6}$, where $\Theta_F$ is an explicit constant depending on $F$, we have: (i) if $(1+\epsilon) \hat{p} \le p = o(1)$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is bipartite; and (ii) if $\frac{1}{n^2} \ll p \le (1-\epsilon) \hat{p}$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is not bipartite. To the best of our knowledge, this work provides the first sharp threshold result obtained for a Turán-type problem in random hypergraphs.

Diffusion processes with reflection, possibly oblique, on the boundary of a given sufficiently regular closed convex domain in Euclidean space, are constructed in a novel way via instantaneous transformations (“foldings") of suitable unconstrained diffusions.

Spatially distributed service systems rely on state-dependent routing to allocate users, tasks, or requests to less-loaded service nodes. In practice, a routing decision does not take effect immediately: the assigned job reaches the selected node only after a lag caused by travel time, communication latency, or actuation. We call this lag the action delay. Whereas delayed-information models treat such a lag as stale information and analyze the model via a delay differential equation, the decision uses the current state and only its execution is deferred, so the job experiencing an action delay must be tracked as part of the state. Representing the action delay by an Erlang phase structure, we obtain a finite-dimensional Markov jump process and build ordinary differential equations that explicitly track the jobs in the delay phase. Exploiting the two-server symmetry, we reduce the dynamics to a difference mode for the server imbalance and derive its characteristic equation for an arbitrary number of phases. This equation coincides with that of the delayed-information model, showing that the two different delays share the same linearized imbalance dynamics. Numerical experiments confirm the fluid approximation and illustrate how the number of phases, the routing sensitivity, and the mean delay govern the transient response of the server imbalance.

The present work investigates the asymptotic behaviours at the zero-noise limit of the first near collision-time and first near collision-location between a pair of independent $d$-dimensional Brownian-driven self-stabilizing (McKean-Vlasov type) diffusions. These asymptotic are considered in a peculiar setting where the systems evolve in a bi-stable landscape and collisions are only triggered by the combined action of the Brownian noises. As the Brownian perturbations fade away, we show that the near collision-time increases at an explicit exponential rate and that related collision-locations persist in specific regions of the space. These results are mainly derived by tailoring classical Freidlin-Wentzell's exit-time (and exit-location) estimates into collision estimates. Similar asymptotic are established for related mean-field interacting particle system approximation, and for the one-dimensional case (where true collisions can be examined

We introduce and study the perturbed $\beta$-corners process, a deformation of the classical $\beta$-corners process. The latter is a probability measure on interlacing arrays of real numbers that, for the classical values $\beta=1,2,4$, describes the joint distribution of eigenvalues of principal submatrices of a Gaussian random matrix with the corresponding GOE/GUE/GSE symmetry. For classical $\beta$, the perturbed process arises from adding a deterministic diagonal matrix $A=diag(a_1,...,a_N)$ to a GOE/GUE/GSE matrix. The eigenvalues of the resulting random matrix depend symmetrically on $a_1,...,a_N$, but this symmetry does not extend to the whole corners process. We extend the construction to all $\beta>0$ via multivariate Bessel functions, and analyze the crystallization ($\beta\to\infty$) of the resulting interlacing array, proving a law of large numbers and a central limit theorem in two regimes. For fixed perturbation, the eigenvalue repulsion dominates, and the array crystallizes on the same lattice of polynomial-derivative roots, with the same discrete Gaussian free field fluctuations, as in the unperturbed case treated by Gorin and Marcus (arXiv:1706.07393). New phenomena appear in the second regime, where the $a_i$'s grow linearly in $\beta$. Then the external source competes with the repulsion at leading order. Here the array freezes on a deformed lattice characterized by a coupled system of optimality equations. The fluctuations are governed by the same discrete Gaussian free field, now attached to the deformed lattice. The deformed lattice equations decouple in the special case of a single spike in the last coordinate. Then the deformed lattice is obtained explicitly by applying one shifted derivative $D_c f = f' + cf$ followed by iterated ordinary derivatives.

We consider level lines of Gaussian free field (GFF) in annulus with alternating boundary conditions. We calculate the probability that all level lines cross the annulus. Such probability is given by the ratio between two partition functions. These two partition functions are constructed via Dubédat's regularized Dirichlet energy. We show that these partition functions are solutions to annulus Belavin-Polyakov-Zamolodchikov (BPZ) equations. In the annulus setup, the number of variables exceeds the number of BPZ equations, so the BPZ system alone does not determine the partition functions uniquely. By establishing sufficiently good control of the two partition functions constructed above, we are nevertheless able to derive the crossing probability.

Estimating the Shannon entropy of discrete distributions with countably infinite support is a challenging problem. In this paper, we investigate the bias of the plug-in estimator $\hat{H}_n$ for the Shannon entropy $H(\boldsymbol{p})$ under an occupancy model whose frequency sequence exhibits regular variation with tail index $\alpha\in(0,1)$. Using Poissonization and the theory of regular variation, we establish the asymptotic relation $|\mathsf{E}[\hat H_n] - H(\boldsymbol{p})| \sim n^{\alpha-1}L(n)C_\alpha$, where $L$ is a slowly varying function and $C_\alpha$ is an explicit constant depending only on $\alpha$ that admits an integral representation. Our result shows that the asymptotic behavior of the bias of the plug-in estimator under power-law frequency distributions is determined by the tail behavior of the underlying distribution.

We prove an improved restricted isometry bound for Gaussian partial circulant matrices with arbitrary prescribed sampling sets. There is a universal constant $C>0$ such that the following holds. Let $1\leq K\leq m\leq N$ be positive integers, let $\Omega\subset\mathbb Z_N$ be any fixed set with $|\Omega|=m$, and let $g\sim\mathcal N(0,I_N)$. For every $\delta,\eta\in(0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\delta$, with probability at least $1-\eta$ over the draw of $g$, provided \[ m\geq C\delta^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/\eta)\}. \] The proof refines the Maurey entropy step in the chaos-process argument by combining a noncommutative Khintchine inequality with a Schatten moment estimate controlled by $m$, replacing one factor $\log(2N)$ in the Krahmer–Mendelson–Rauhut bound by $\log(em)$.

This paper computes Poisson space noise functionals, $P^{\prime}(u)$, realised in a Gel'fand triple built from a Lévy measure $\lambda_{\beta}(u)du$. We isolate three discretisation parameters: a small-amplitude cut-off, a Donsker delta truncation M, and a chaos order N. For a stable-type intensity $\lambda_{\beta}(u)=cu^{-1-\alpha}$ ($0<\alpha<2$), replacing discarded small amplitudes with matched Gaussian space noise improves the Wasserstein-1 error from $O(\epsilon^{1-\alpha/2})$ to $O(\epsilon)$. The residual is asymptotically normal at $O(\epsilon^{\alpha/2})$. This compensation reduces computational complexity from $O(\tau^{-2\alpha/(2-\alpha)})$ to $O(\tau^{-\alpha})$. We also evaluate the Gamma-type boundary ($\alpha=0$) and exponential tempering. Truncations converge algebraically (M) and super-geometrically (N). All predicted rates are tightly confirmed by deterministic numerical experiments via Gil-Pelaez inversion, eliminating Monte Carlo noise.

Fix $p\ge 3$. We establish a separation between regularized bulk universality and unrestricted annealed complexity for the pure spherical $p$-spin Hamiltonian with independent non-Gaussian tensor coordinates. There is a symmetric, compactly supported disorder law with a smooth density, matching the first $2p$ Gaussian moments, for which the expectation of a positive, frame-averaged regularization of the one-point Kac-Rice functional is asymptotic to its Gaussian counterpart, while the unrestricted critical-point count in a compact energy window has a lower exponential rate strictly above the Gaussian limit. The obstruction is a coherent block involving on the order of $N^{1/p}$ coordinates, so neither finite moment matching nor a uniform entry bound restores unregularized annealed universality. For uniformly subexponential disorder matching the first $m$ Gaussian moments, the regularized functional on incoherent frames satisfies $\left|\log\left(\mathbb{E}\mathcal{Z}_{N,L}^J/\mathbb{E}\mathcal{Z}_{N,L}^G\right)\right|\le C N q_N^{m-1}$, where $q_N=L^{p-1}N^{-(p-2)/2}(\log N)^{(p-1)/2}$. Thus mean and variance matching imply regularized pressure universality for every $p\ge 3$, and the expectation ratio tends to one when $(m-1)(p-2)>2$. We identify the Gaussian variational limit and prove the Gaussian quadratic energy-excursion upper bound uniformly over profiles asymptotically supported on $o(N)$ coordinates under a Gaussian-rate profile-tail condition. Finally, we give a conditional reduction to exact universality: once two one-sided de-regularization defects vanish, an exact max formula makes control of the localized complement necessary and sufficient. The defects vanish in the Gaussian model.

In this paper, we establish asymptotics for the small value probabilities of additive and derivative martingales in both supercritical branching Brownian motions and super Brownian motions, thereby extending the corresponding results for Galton–Watson processes and continuous-state branching processes. For the derivative martingale in branching Brownian motion, our result also agrees with the findings in the arXiv version of Arguin et al. [arXiv:1008.4386 v1] and with those of Hu [Ann. Inst. H. Poincaré Probab. Stat., 2016].

In high-dimensional hyperbolic space, concentration of a radial measure near a shell need not determine the pyramid limit: angular concentration and hyperbolic expansion alter separation. An effective radius is the scale on which positive-mass sets actually separate, which the raw radius need not give. With vanishing rescaling and radial fluctuations, radius convergence gives weak convergence to the pyramid of all metric measure spaces with the resulting diameter bound. At modal radial Gibbs shells, exponential decay of intrinsic shell curvature and dimension-normalized tangential Bakry-Émery Ricci curvature recovers the radius. The Gaussian intrinsic to hyperbolic volume and that obtained by wrapping a Euclidean Gaussian have different critical orders. They are Lévy below those orders, infinitely dissipate above them, and at criticality converge to the corresponding diameter-bounded pyramids.

This article studies the conditional-density equation and its pathwise transformation in local stochastic rough volatility models, with rough Heston (rHeston) as the main explicit example. Under the stated common-filtration, measurability, predictability and spatial-regularity assumptions, we show that the Ito-Wentzell random-PDE reduction of the conditional density SPDE remains valid under local stochastic rough volatility. After fixing a common-environment realization and the associated stochastic flow, the transformed equation becomes a deterministic PDE with path-dependent coefficients. This yields a pathwise Fokker–Planck formulation that connects naturally with Rao–Blackwellized calibration. In the pure rough Heston case, the transformed coefficients simplify and the conditional density admits an explicit lognormal form.