Papers

Gaussian process (GP) modeling is widely used in computational science and engineering. However, fitting a GP to high-dimensional inputs remains challenging due to the curse of dimensionality. While various methods have been proposed to reduce input dimensionality, they typically follow a two-stage approach, performing dimension reduction and GP fitting separately. We introduce a Bayesian framework that seamlessly integrates dimensionality reduction with GP modeling and inference. Our approach, built on a hierarchical Bayesian model with priors on the Stiefel manifold, enforces orthonormality on the projection matrix and enables posterior inference via Hamiltonian Monte Carlo with geodesic flow. Additionally, we extend this framework by incorporating Deep Gaussian Processes (DGP) with built-in dimension reduction, providing a more flexible and powerful tool for complex datasets. Through extensive numerical studies, we demonstrate that while the proposed Bayesian method incurs higher computational costs, it improves predictive performance and uncertainty quantification, providing a principled and robust alternative to existing methods.

In this work, we investigate the numerical approximation of degenerate Langevin-type stochastic partial differential equations (SPDEs) in two spatial dimensions. These SPDEs arise in stochastic dynamics and mathematical finance, among other applications. In order to handle the mixed deterministic-stochastic structure of the equation and the degeneracy of the differential operator, we propose a semi-implicit Milstein finite difference scheme for the numerical solution. Through the Fourier analysis of the mean-square stability and convergence, we derive explicit conditions on the coefficients under which the scheme is stable, jointly with explicit convergence rates in terms of the discretization parameters. We further embed the proposed scheme within a Multilevel Monte Carlo (MLMC) framework to reduce the computational cost associated with SPDE simulations, and we derive its theoretical computational complexity. Numerical experiments confirm theoretical convergence rates and show that the MLMC strategy achieves an accuracy comparable to standard Monte Carlo at a fraction of the computational cost, reducing the complexity from $\mathcal{O}(\varepsilon^{-5})$ to $\mathcal{O}(\varepsilon^{-3})$ for a target root-mean-square error $\varepsilon$. These results show that combining semi-implicit Milstein schemes with MLMC techniques provides an effective approach for the numerical simulation of Langevin-type SPDEs.

This paper studies the convergence of the Sinkhorn algorithm for the Schrödinger bridge problem with regime switching, as introduced in Zlotchevski and Chen (2025). We consider a class of regime-switching stochastic systems on the hybrid state space $E=\mathbb{R}^d\times\{1,\ldots,m\}$, and construct the Sinkhorn iteration through the associated equivalent entropic optimal transport formulation. The main result of this paper is the exponential convergence of the Sinkhorn algorithm in relative entropy under compactness assumptions. Our proofs are inspired by the arguments recently developed for proving exponential convergence of the classical Schrödinger problem in Chiarini, Conforti, Greco and Tamanini (2024), Eckstein (2025). We perform a similar analysis for the partially observed terminal setting, where only the marginal distribution of the continuous component is prescribed at the terminal time, while the discrete regime is unobserved.

The paper introduces a stochastic model for the spontaneous formation of traffic waves on a highway. This is formulated in terms of a conservation law with discontinuous, gradient-dependent flux. In an unstable regime, the non-uniqueness of solutions allows for the emergence of $N$-shaped spikes in the traffic density, at random points and times. Bounds are proved on the expected value of the total variation of the random solution and on the expected number of shocks. Further bounds are obtained on the average velocity and on the expected average acceleration of cars, along a given stretch of highway. Finally, it is proved that the Markov process, whose paths are random solutions to the conservation law, admits a unique stationary probability distribution and is ergodic.

A stochastic representation and sampling algorithm for Morillas-type copula-to-copula transformations and related distortions of multivariate distribution functions is derived, resulting as a byproduct in a novel sampling scheme for Archimedean and Archimax copulas. This closes a methodological gap and facilitates simulation-based applications of distorted copulas. For stable tail dependence functions (stdfs), a Morillas-type distortion framework is introduced, where monomial distortions with exponents below 1 are shown to preserve stdfs via a domain-restricted Pexider equation analysis. This characterization is leveraged to identify distortions preserving extreme value copulas, and convex combinations of distorted stdfs are proposed to increase flexibility in extremal dependence modeling. The impact of distortions on maximum domain of attraction limits is also analyzed. Explicit limiting EVC distortions are identified under non-restrictive regular variation assumptions. Examples of absolutely monotone distortions allowing to fine-tune the extreme value behavior after distortion, but also of non-regularly varying 2-absolutely monotone distortions are given.

We introduce anisotropic Banach spaces of distributional 1-forms on compact surfaces designed to capture the fine regularity properties of Morse gauge-fixed Yang–Mills random connections. Using singular connections supported on unstable curves of a Morse gradient flow, we provide a new description of the moduli space of flat connections and of its canonical Atiyah–Bott–Goldman symplectic form. When the group is the special unitary group, we prove that in the zero-area limit, the Yang–Mills random connection converges, within these anisotropic spaces, to a random distributional 1-form whose law coincides with our Morse-theoretic representative of the Atiyah–Bott–Goldman measure. Our approach extends the works of Witten, Forman, Liu, and Sengupta by establishing the zero-area limit of the Yang–Mills measure at the level of random distributional connections, rather than only at the level of holonomies. This answers a question of Thierry Lévy on the semiclassical limits of Yang–Mills random connections.

Motivated by sampling and approximation schemes arising in nonsmooth optimization, we study the stability of parameterized set-valued integrals under weak perturbations of the underlying probability distribution. For a compact parameter set, we show that compact convex-valued and jointly outer semicontinuous integrands induce set-valued integral maps that converge graphically in excess distance along any weakly convergent sequence of probability measures. The result holds under a superlinear integrability condition and provides a unified stability principle for measure approximations of set-valued expectations. We discuss the sharpness of the assumptions through examples. In particular, we emphasize that joint outer semicontinuity is required in general and that the superlinear envelope condition is tight relative to the classical i.i.d. empirical setting. As a consequence, we obtain outer stability of solution sets for stochastic generalized equations. We illustrate the stability result in several settings, including stochastic nonsmooth optimization with Markovian sampling, smoothing by mollifiers, and parameter-dependent distributional dynamics.

This paper presents a unified cost-modeling and optimization framework designed to evaluate hybrid discrete-variable/continuous-variable quantum key distribution (DV/CV-QKD) infrastructures operating within the SAFE (Secure and Efficient) long-term storage (LTSS) protocol. Our methodology jointly addresses the information-theoretic and computational dimensions of cryptographic durability over multi-decade horizons by coupling a quantitative risk model with a stochastic integer linear programming (ILP) formulation. Going beyond idealized physics-informed limits, the framework incorporates realistic next-generation industrial implementations and multiplexed coexistence specifications, together with a sample-average approximation (SAA) pipeline, to determine cost-optimal and structurally feasible SAFE topologies under transmission-budget and security constraints. The proposed framework is evaluated on both randomized synthetic deployments and realistic metropolitan-scale QKD infrastructures, including the Paris Metro-Scale and Greater Paris networks, to characterize the interplay between network topology, QKD modality, and deployment cost while enforcing a target global compromise tolerance $\epsilon$. Numerical results show that the economically optimal architecture is highly scenario-dependent and that both hybrid and homogeneous modality allocations naturally emerge from the optimization. More importantly, the analysis reveals a non-monotonic relationship between the minimum required QKD link capacity and the infrastructure cost: contrary to intuition, increasing the minimum required per-link secret-key rate may reduce the overall deployment cost by enabling a global transition toward more economical CV-based solutions, whereas further capacity increases do not necessarily yield additional savings.

We introduce a microscopic framework that gives the first rigorous derivation of the Schwinger–Dyson thermodynamics of the Sachdev–Ye–Kitaev model. For fixed, even interaction order $q\geq 4$, we prove that the annealed and quenched normalized pressures converge to the Schwinger–Dyson pressure $p_{{\rm SD},q}(\beta)$, at any fixed inverse temperature $\beta>0$. The proof is derived from the finite-$N$ Gibbs state, and it contains three new ingredients: a single-site cavity expansion that keeps the bulk Gibbs state intact, a finite-dimensional locality estimate yielding label-uniform conditional factorization of the Euclidean cavity fields, and an exact Majorana-bath representation of the leading diagrams. We further determine the asymptotic location of the largest eigenvalue, which answers a question posed by Feng–Tian–Wei. Consequently, the sample free-energy density converges almost surely to the zero-temperature value along every sequence $\beta\equiv \beta_N\to\infty$.

We study connections between discrete probability distributions and the statistical mechanics of small systems. Using probability generating functions, we develop the theory of power series, infinitely divisible, and scalable distributions of non-negative integer-valued random variables, and introduce the class of Markovian distributions that arise naturally in stationary solutions of birth-death processes and in scalable infinitely divisible distributions. These results are applied to the grand canonical ensemble description in statistical mechanics: the infinite divisibility leads to a quasiparticle picture of an interacting gas, and the virial expansion is linked to the combinants of the distribution. A kinetic model of the liquid-vapor phase transition is presented, in which the particle-number distribution at the critical point converges to the Discrete Stable distribution - the fixed point of a renormalization semi-group transformation. We also show that the scalability of the particle-number distribution is preserved in Tsallis non-extensive thermodynamics, even though infinite divisibility fails. Moreover, deformed discrete distributions, including the negative binomial as a q-deformation of the Poisson, arise naturally in this setting.

We prove strong convergence rates for tamed Euler schemes of multidimensional stochastic differential equations with singular drift, multiplicative Brownian noise, and multiplicative Levy noise. The drift is assumed to satisfy a Ladyzhenskaya-Prodi-Serrin type condition, while the diffusion and jump coefficients have Sobolev-type spatial regularity. The proof is based on a nonlocal Zvonkin transform, Krylov-type estimates, and a stochastic Gronwall argument. The main novelty is an error decomposition that isolates two jump-induced contributions: a compensated jump martingale error and a nonlocal compensator error caused by Euler freezing. This decomposition yields explicit rates and recovers the Brownian-type estimate when the jump coefficient vanishes.

In this note, we introduce a generalisation of multiplicative chaos measures which is both non-Gaussian and non-Abelian. The renormalisation procedure takes as inputs an irreducible unitary representation of a compact connected Lie group, together with a Kac-Moody unitarising measure at some level $\kappa$, and outputs a random measure on the circle with values in the space of positive definite Hermitian endomorphisms of the representation space. So far, our construction is valid for the range of $\kappa$-values corresponding to the $L^2$-phase. The proof follows the usual route in the theory of multiplicative chaos, relying on an exact formula for the one-point function and a bound on the two-point function at colliding points. These expressions are derived using the rich algebraic structure of the theory: namely, we establish a one-dimensional version of the Knizhnik-Zamolodchikov equations.

We study averaging-learning dynamics without an exogenous ground truth: the reference signal is generated endogenously by the population. The dynamics combine a time-varying averaging matrix, a learning-source matrix, and a learning matrix, typically diagonal. Dobrushin-type contraction controls the decay of oscillations and yields asymptotic agreement. Under a summability condition, the backward products converge exponentially to rank-one limits. With summable perturbations, the process converges to a random consensus state. For i.i.d. perturbations, we prove a central limit theorem for the centered process: the limiting Gaussian law is supported on the agreement direction. Thus, despite the multi-agent dynamics, the long-time fluctuations are asymptotically one-dimensional. We also record a pairwise Dobrushin formulation that clarifies the dynamic agreement-class geometry underlying the one-class regime.

Given observations $\mathbf x=(x_1,\dots,x_n)$, Gaffke (2005) defined \[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \] and conjectured that it is a $p$-value whenever the inputs are independent e-values. Recently, Vlassis and Thomas (2026) proved this conjecture. Inverting the tests for observations in $[0,1]$ gives the confidence interval studied by Learned-Miller and Thomas (2020), which reduces to Clopper–Pearson for Bernoulli data. We give a finite- and large-sample account of Gaffke's test and interval. First, for every $\mathbf x\in[0,\infty)^n$ and every elementary symmetric polynomial $e_k$, \( K_n(\mathbf x)e_k(\mathbf x)\le {n\choose k}, \) so the Gaffke $p$-value never larger than the SymPol $p$-value of Ming et al. (2026). However, Gaffke's p-value is inadmissible. For $n=2$, we construct a valid rule that is strictly smaller on mixed configurations and is the unique admissible rule that dominates $K_2$. A neutral-face extension proves inadmissibility of $K_n$ for every $n\ge2$. If one independent uniform random variable is allowed, there is an even simpler full-dimensional improvement: on the upper orthant, where $K_n(\mathbf x)=1/\prod_i x_i$, replace it by $U/\prod_i x_i$. The equal-tail Gaffke confidence interval $I_n$ is nevertheless first-order asymptotically efficient: for iid observations on $[0,1]$ with unknown variance $\sigma^2>0$, \[ \sqrt n\,\operatorname{Width}(I_n)\longrightarrow 2\sigma z_{1-\alpha/2}\qquad\text{almost surely}. \] Our simulations also find that, among a variety of bounded-mean intervals considered, the Gaffke interval is the shortest, including comparisons with a recent empirical Berry–Esseen procedure having the same first-order Gaussian target.

In this article we establish global a priori estimates on the solution of a generic rough differential equation driven by an $\alpha$-Hölder path covering the full range of regularity $\alpha\in(0,1)$, under the hypothesis of Lipschitz continuity (but neither boundedness nor coercivity) of specific combinations of the coefficients and their derivatives, called elementary differentials. Our main tool is a closed formula for the remainder of a generalised Taylor expansion in terms of products of gradients of the elementary differentials, which allows to take advantage of cancellations in an optimal way.

Let $f$ be the density of the Chernoff random variable $\mathrm{argmax}_{t\in\mathbb{R}}\{W(t)-t^2\}$, where $W$ is a two-sided Brownian motion. This note proves the conjecture of Balabdaoui and Wellner (2014) that $f$ is strongly log-concave. The proof was generated in its entirety by GPT-5.6 Sol.

Let $\mathcal{X}$ be a finite set, let $p$ and $Q$ be strictly positive probability distributions on $\mathcal{X}$, and let $u\in\mathbb{R}^{\mathcal{X}}$ be a log-weight. Define $\Psi_p(u)=\log\sum_x p(x)e^{u(x)}$ and $\pi_{p,u}(x)=p(x)e^{u(x)-\Psi_p(u)}$. The variational free energy $F(Q;p,u)=D(Q\|p)-\mathbb{E}_Q[u]$ satisfies the exact identity $\Psi_p(u)=-F(Q;p,u)+D(Q\|\pi_{p,u})$, whose residual is the canonical divergence of the dually flat simplex. We prove a converse. Suppose $G(Q;p,u)=A(Q,p)+B(Q,u)$ is additively separable and its gap $\Psi_p(u)+G(Q;p,u)$ is nonnegative and vanishes whenever $Q=\pi_{p,u}$. Then $G=F$ and the gap equals $D(Q\|\pi_{p,u})$; $A$ and $B$ are unique up to a $Q$-dependent additive gauge. The gap is not assumed to be a divergence or to depend only on the target, and no continuity, measurability, differentiability, convexity, or membership in a prescribed divergence family is assumed. The proof bounds increments of a corrected accuracy term by centered log-moment-generating costs of order $k^{-2}$ per step. Telescoping $k$ equal exponential tilts gives an $O(k^{-1})$ bound and forces every increment to vanish. Thus target factorization and strictness are conclusions. Counterexamples show that separability, nonnegativity, and tightness are each necessary. Within this class, no $\alpha$-divergence distinct from Kullback–Leibler and no Rényi divergence of order other than one can occur as the residual.

We study doubly reflected backward stochastic differential equations with jumps and two completely separated right-continuous with left limits barriers in a filtration generated by an inhomogeneous Levy process. We establish existence and uniqueness results under a stochastic Lipschitz condition on the driver by means of a penalization method. We also prove a comparison principle and present two closely related applications. The first concerns the nonlinear valuation of an American game option in such a Levy market, while the second addresses the associated generalized Dynkin game under nonlinear expectation. Moreover, under suitable semicontinuity assumptions on the barriers, we establish the existence of a saddle point for the game.

We construct a vector space whose defining characteristics are rooted in univariate regular variation of random variables. Specifically, the base vector space $\mathbb{V}_b$ consists of random variables whose limiting tail probabilities, when scaled by regularly varying functions of the form $b(s)=s^\alpha L(s)$, are finite. Defining a subspace ${\cal N}_b$ corresponding to random variables in $\mathbb{V}_b$ whose limiting tail probabilities are zero when normalized by $b(s)$ allows the base space $\mathbb{V}_b$ to be partitioned into equivalence classes. We define a vector space $\mathbb{W}_b$ consisting of these equivalence classes, and show its nonzero elements are equivalence classes of regularly varying random variables. We show that a natural norm exists for $\mathbb{W}_b$ if $\alpha > 1$. We show that the equivalence classes and convergence in norm are different than more familiar vector spaces of random variables. Turning our attention to extreme value modeling, we consider finite-dimensional subspaces of $\mathbb{W}_b$ whose basis vectors are jointly regularly varying. We show that in the case $\alpha = 2$, the previously defined tail pairwise dependence measure serves as an inner product. As any finite-dimensional space is complete, we can use the projection theorem to perform linear prediction.

We derive complete asymptotic expansions for the binomial mass at a bounded lattice displacement and for the mean absolute deviation $E|X-Np|$, $X\sim Bin(N,p)$, with $0<p<1$. De Moivre's exact formula reduces the latter problem to the local mass at $\nu=\lceil Np\rceil$, so the coefficients depend on the oscillating displacement $h_N=\lceil Np\rceil-Np$. We show that the full expansion is governed by Bernoulli polynomials evaluated at this displacement; equivalently, the lattice correction is an Appell shift in the Stirling series. The calculation is based on a gamma-quotient expansion with unequal linear scalings, stated with uniformity in the bounded shift. In passing from the local mass to the mean absolute deviation, the elementary, non-Bernoulli part of the shift cancels term by term against the De Moivre prefactor, leaving coefficients that are pure Bernoulli polynomials. As consequences, the classical first correction of Frame and Johnson is embedded in the general coefficient sequence, and the Cesàro means of the oscillating coefficients are obtained from the multiplication theorem for Bernoulli polynomials. Finally, although an oscillating asymptotic expansion does not bound its function by truncation, De Moivre's identity together with Robbins's form of Stirling's formula yields an elementary two-sided bound of logarithmic width $O(N^{-2})$ in the interior; and at an integer mean the logarithmic expansion reduces to a sign-alternating series in odd powers of $N^{-1}$ which we prove to be enveloping, via a sign-definite Binet-kernel representation of the combined three-gamma Stirling remainder: successive truncations bracket the mean absolute deviation.

We develop an exact finite-population stochastic framework for SIR epidemics evolving under Markovian switching between intervention regimes. The epidemic state is augmented by a finite phase component, allowing transmission, recovery, and direct immunity-acquisition rates to depend on the active regime. Phase-transition intensities may depend on the current epidemic state, so that policy escalation can react to the number of infectious individuals. Exploiting the monotonicity of the susceptible compartment, we derive level-wise recursions for the joint Laplace–Stieltjes transform and probability generating function of the extinction time and the number of infections generated before extinction. These recursions yield the infection-count distribution, conditional extinction-time transforms, and mixed moments linking epidemic duration and infection burden, while replacing a large global linear system with small phase-level solves. The framework is illustrated using weekly mpox incidence data from Luxembourg. A baseline one-phase SIR model is calibrated by maximum likelihood under a Poisson observation model. The calibrated baseline is then used for conditional comparisons of fixed control regimes, early versus delayed strict intervention, vaccination-supported control, and state-dependent escalation. The results show how switching mechanisms affect both the total number of infected individuals and the extinction time, including their dispersion. Since the switching mechanisms are specified rather than estimated from the intervention history, the results are conditional model-based comparisons rather than estimates of the historical effects of interventions in Luxembourg.

We consider the problem of finding the root vertex of a random uniform attachment tree, when the union of the unlabeled tree and an Erdős-Rényi random graph $\mathbb{G}(n,p)$ is observed. We prove that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that depends only on $\varepsilon$ and not on $n$, such that it contains the root with probability at least $1-\varepsilon$. This affirms a conjecture of Crane and Xu (2021). Our approach ranks vertices by their Jordan centrality in the largest component of the subgraph spanned by high-degree vertices. We show that the same approach works in other noise models as well.

Disordered systems such as spin glasses and polymers characteristically exhibit random energy landscapes with many macroscopically separated energetic valleys corresponding to near-ground states. This high complexity renders these systems extremely sensitive to perturbations of external parameters. For instance, the support of associated Gibbs measures may change macroscopically under such perturbations, a phenomenon known as chaos in the literature. In experiments, chaotic phenomena are typically studied via temperature perturbations. In this article, we initiate the rigorous study of temperature-chaotic properties of the continuum directed random polymer (CDRP), a canonical model in the KPZ universality class. The CDRP is driven by white noise and is parametrized by inverse temperature $\beta$, and is known [Wu '26, Das-Zhu '24] to converge in the zero-temperature limit $\beta \to \infty$ to the directed landscape constructed in [Dauvergne-Ortmann-Virág '22], the putative universal scaling limit of models in the KPZ universality class. The main result of this article considers the CDRP free energies coupled through the same white noise at a pair of inverse temperatures $(\beta_1, \beta_2)$, and shows that they decouple in the limit $\beta_2 \gg \beta_1 \gg 1$, converging to a pair of independent directed landscapes. This is the first such "energetic de-correlation across temperatures" result. Our key estimate measures the "pivotality" or "influence" of spatially thin strips in models of last passage percolation. As a byproduct, the proof strategy also allows to show that the directed landscape is a two-dimensional black noise (in the sense of [Tsirelson-Vershik '98]), previously conjectured by Virág. This provides the third known example of a two-dimensional black noise after critical planar percolation [Schramm-Smirnov '11] and the Brownian web [Ellis-Feldheim '16].

In this work, we study small-noise asymptotics of risk-sensitive exit-time control problems governed by stochastic differential equations with path-dependent coefficients. Our main result establishes the convergence of the $\log$-transformed exit-time problem to a deterministic control problem with path-dependent coefficients. For its proof, we first derive a novel variational representation for general $\log$-transformed stochastic control problems with path-dependent coefficients, combining tools from the theory of path-dependent partial differential equations and convex expectations on path spaces. In a second step, we use probabilistic methods to analyze the convergence of the resulting variational formulas. To illustrate the scope of our analysis, we consider a computable example for a stochastic differential equation with memory and characterize the limiting problem and associated control strategies.

We study the small-regularization limit for $L^p$-regularized optimal transport with $1<p<\infty$ and for entropically regularized optimal transport (EOT). The exact first-order (respectively, second-order) asymptotics are determined explicitly under mild assumptions on the source and target measures. Our work generalizes the existing results for quadratic and entropic regularization, and connects them by a natural interpolation via $p\in(1,2)$. We derive all these asymptotics in a unified manner by a novel approach that separates the local computation of the optimal profile from the global enforcement of the marginal constraints: convex duality leads to Gaussian profiles for entropy and Barenblatt profiles for $L^p$-regularization, while a quantization construction turns these local profiles into couplings.

We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alpöge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass–Connell–Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.

Entropy regularization smooths equilibrium policies in time-inconsistent stochastic control. At low temperature, the same Gibbs response can strongly amplify errors in learned rewards and dynamics. We show that the derivative of an exploratory equilibrium is governed by a backward Volterra-parabolic resolvent. Along an aligned positive mode, a lower bound has the same exponential order. A block decomposition identifies the source of the amplification: causal paths contribute powers of 1/tau, whereas a positive feedback cycle can produce exponential growth. At fixed temperature, a local equilibrium branch is twice differentiable with respect to finite-dimensional model parameters, which yields a function-valued delta method. A bounded uniformly elliptic diffusion realizes this path-cycle distinction in every finite dimension. Closing one positive cycle changes the root-n linear-response boundary from a power law to order 1/log n; along the cyclic Perron mode, right-endpoint discretization is relatively consistent exactly when N tau^2 -> infinity. An affine model also gives an exact nonlinear transition at the Lambert-W temperature beta T / W(beta T sqrt(n)). Numerical calculations illustrate these rates.

We construct the periodic directed landscape, which is the conjectured scaling limit for periodic models in the Kardar-Parisi-Zhang universality class. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape. Moreover, we confirm conjectures by Baik, Liao and Liu on the local structure of the periodic KPZ fixed point, and establish the convergence of periodic ASEPs to periodic KPZ fixed points, coupled according to the same periodic directed landscape. Our main tool to construct the periodic directed landscape (and prove convergence to it) is a technique for gluing full-space directed landscapes (and their prelimits), which is of independent interest.

We introduce a family of random connected closed subsets of planar Brownian motion, called Brownian loop-catchers, which interpolate between the continuum loop-erased random-walk (LERW) and the Brownian trace. This provides the canonical continuation of Brownian loop soup clusters to central charges $-2\le c<0$. For each such $c$, the corresponding loop-catcher satisfies the following recovery property: adding all loops from an independent Brownian loop soup of intensity $-c/2$ that intersect it recovers the Brownian trace. Furthermore, no such law exists for $c<-2$. We also show that its outer boundary is locally SLE$_\kappa$ with $\kappa = \frac{1}{3}\left(13 - c - \sqrt{(1-c)(25-c)}\right)\in[2,\frac83)$, and the probability that it intersects an interior ball of radius $\varepsilon$ is asymptotically proportional to $|\log\varepsilon|^{-1+\frac{c}{2}}$ when $-2<c<0$. Therefore, a planar Brownian trace contains an SLE$_\kappa$-type curve for every $\kappa\in[2,\frac83]$. Our construction begins with a random-walk loop-catcher on any finite graph such that recursively inserting loops from an independent random-walk loop soup to it recovers the original random walk. The key ingredient is a new entangled multipath LERW, which recovers a union of independent random-walk paths when decorated with a single common random-walk loop soup. We then prove that the random-walk loop-catcher converges to the Brownian loop-catcher under lattice approximations. To this end, we propose a novel Green function test which converts the recovery property of the Brownian loop-catcher into all mixed moments of Green functions in the remaining domain, based on the entangled multipath LERW. Consequently, the recovery property characterizes the full law of the Brownian loop-catcher, not only its filling. The Green function test also extends to the three-dimensional case.

We prove full replica symmetry breaking for the zero-field Sherrington-Kirkpatrick model at zero temperature: the Parisi minimizer is absolutely continuous, has a smooth density, and has support $[0,1)$. At inverse temperature $\beta>1$, [arxiv.org/abs/2607.11756v3] recently proved that the Parisi measure has support $[0,q_\beta]$. Here, we show $q_\beta$ converges to $1$ as $\beta\to\infty$.