Papers

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

We develop a two-dimensional Allen-Cahn theory for interfaces with boundary in the line-bundle framework introduced by Fröhlich and Struwe. The central new object is a model solution on the punctured plane whose nodal set is a half-line. Near the boundary of an interface, this solution plays the role of the heteroclinic profile in the classical interior theory. However, unlike the interior setting, where the analysis in directions normal to the interface reduces to the ODE satisfied by the heteroclinic, the boundary model has non-flat level sets and must be studied as a genuinely two-dimensional elliptic solution. In the first part of this paper, we construct this model solution and develop its stability and invertibility theory. In the second part of this paper, we present the main application: we construct Allen–Cahn sections whose nodal sets concentrate on any prescribed finite collection of disjoint line segments in the plane. The construction uses a Lyapunov-Schmidt reduction, but the boundary introduces a new difficulty: it creates large error terms along the interior of the interface. Controlling these terms requires a refined ansatz and new gluing arguments beyond the standard interior Allen-Cahn theory.

Surface reconstruction from point clouds is a fundamental problem in computational geometry with broad applications in computer graphics, medical imaging, and manufacturing. In this paper, we propose an efficient method for solving a curvature-regularized surface reconstruction model. To avoid the computational cost associated with reinitialization in traditional level set methods, we represent the reconstructed surface by an indicator function. By introducing an auxiliary variable, we reformulate the original optimization problem as the computation of the steady-state solution of an initial value problem. We then develop an operator-splitting method to decompose the resulting problem into two tractable subproblems, one of which can be efficiently solved by iterative thresholding. The proposed approach combines the advantages of curvature regularization, operator splitting, and threshold dynamics. Numerical experiments show that the proposed method is computationally efficient and can accurately reconstruct sharp corners and concave features.

For a given real matrix $A\in\R^{n\times n}$, a Pareto eigenvalue of $A$ is a real number $\lambda\in\R$ for which there exists a nonzero vector $x\in\R^n\setminus\{0\}$ such that $$ 0\leq x\perp Ax-\lambda x\geq0. $$ We prove that every matrix $A\in\R^{4\times4}$ has at most $23$ distinct Pareto eigenvalues. We first prove the result for matrices of order $4$ satisfying two conditions: every real eigenvalue of a principal submatrix is simple, and two different principal submatrices have no real eigenvalue in common. For these matrices, a fixed point argument shows that the number of Pareto eigenvalues is odd. Previous known results show that the Pareto capacity of order $4$ is between $23$ and $26$. Thus only $25$ remains to exclude. We exclude this case by using the support profiles in order $3$ and identities involving eigenvectors of principal submatrices. A perturbation and fixed point index argument then extends the bound to all real matrices of order $4$. An exact symbolic computation certifies that an explicit matrix of order $4$ has exactly $23$ distinct regular Pareto eigenvalues.

We study a restriction of the classical Erdős–Sós problem, the extremal number of trees, to the class of bipartite host graphs, both when only the order of the host is prescribed and when its two part-sizes are fixed. We give natural lower-bound constructions and formulate corresponding linear upper-bound conjectures. We apply a weighted variant of $k$-minimality to prove upper bounds for a broad family of trees including brooms, trees with part-sizes obeying certain inequalities, and all trees on at most seven vertices, resolving part of a problem of Caro, Patkós and Tuza up to additive constants. We also relate the fixed-part extremal number of a tree to the ordinary extremal number, and consider an oriented bipartite extremal function analogous to the Zarankiewicz function.

We investigate the strong approximation of scalar stochastic differential equations when the available standard information about the drift coefficient, the diffusion coefficient, the derivative of the diffusion coefficient, and the observed Wiener path is corrupted by noise. The precision of the drift, diffusion-information, and Wiener-path observations is described by three nonnegative parameters $\delta_1,\delta_2,\delta_3$, where $\delta_2$ controls both the noisy diffusion coefficient and the separate noisy derivative oracle required in the Milstein correction. We analyze a randomized Milstein scheme based only on this noisy information and prove, for $r\geq 2$, that its $L^r$-error is bounded by $C(n^{-\min\{\gamma_1+1/2,\gamma_2\}}+\delta_1+\delta_2+\delta_3)$, where $n$ is the number of time steps and $\gamma_1,\gamma_2$ are the temporal Hölder exponents of the coefficients. We also prove a matching minimax lower bound in the randomized standard-information model considered in the paper. In particular, the Wiener-path contribution proportional to $\delta_3$ is unavoidable, and the noisy randomized Milstein scheme is minimax order-optimal.

An annulment structure consists of a numbered domain of distinctions with a $\Sigma^0_1$ manifestation predicate; an adjudicator is a partial map assigning to distinctions the verdicts annulled or exempt. We ask which domains admit an adjudicator that is total, correct, and complete in its mission – annulling every non-manifesting member – and locate the boundary exactly. On the positive side, domains with decidable manifestation admit canonical adjudicators with certificates, instantiated for Presburger arithmetic and finite-state re-entry systems. On the negative side, adjoining a single nullary gate, by which a distinction may query the verdict passed on itself, destroys decidability uniformly: the extended domain carries a trichotomy in which every adjudicator fails totality, exhaustiveness, or soundness at one distinction fixed in advance. Bounding the depth of self-address refines this: each finite level remains decidable, the hierarchy of iterated verdicts is the synchronous update of a Boolean network on the query graph, deciding stabilisation is PSpace-complete for explicitly presented networks, and periods as large as $2^n-1$ occur at closure size $n$. In the limit what fails is convergence, not decidability; the diagonal distinctions oscillate with period two, and their mean frequency of $1/2$ is the value a reflective oracle is forced to return there. Over the standard domain relative to an oracle $X$, the least Turing degree of an adjudicator with all three properties is the degree of $X'$: adjudication costs one jump per level. The boundary is a three-way trade among determinacy, correctness, and residence at the level adjudicated. The fixed-point core holds over every precomplete numbering in the sense of Ershov. The framework falls on the intensional side of the divide between Kleene's two recursion theorems. Gödel's incompleteness theorems are nowhere used.

We prove the existence of strongly converging unitary representations in various new settings of countable groups, in particular, arising as extensions by exact groups: semidirect products with amenable groups; generalized wreath products with abelian base; graph wreath products; various free-by-cyclic groups including Gersten's group; and general Bernoulli shift crossed products on $C^*$-algebras.

Let $\ell(n)$ denote the least non-trivial reduced quadratic residue modulo $n$; that is, $\ell(n)$ denotes the smallest square-free integer $r>1$ with $(r,n)=1$ and $r\equiv x^2 \bmod {n}$. We establish nearly optimal bounds for $\ell(n)$, both in terms of the magnitude of $n$ and of its number of prime factors $\omega(n)$. In particular, we construct moduli $n$ for which $\ell(n)$ is unexpectedly large. As an application of our results, we prove bounds for the rate at which binary quadratic forms with bounded discriminant represent all positive integers up to $N$.

Promoting cooperative behaviour amongst decision makers has key implications for the long term sustainability of social systems. Incentives can promote cooperation in situations where defection is more favourable. Previous research has identified optimal decentralised incentives in structured populations where the social environment is described by a generic single-layer networks. Optimality is here intended in the sense of cost minimisation. Here, we provide and optimal incentive strategy for structured populations in general network, a rather unexplored case. Further, we look at populations where, through the network structure, each player interact with one set of neighbours while contributing to an opinion diffusion dynamics from a second set of neighbours. Hence we cover the case of multilayer networks. To fill this gap, we provide a solution to the optimal control problem by solving the Hamilton-Jacobi-Bellman equation and derive an analytic solution for distributing incentives in multilayer networks. By implementing the dynamics of the networked Prisoner's Dilemma Game, we provide a feedback control loop that yields an optimal incentive distribution over time. The dynamic incentive depends on the current state of cooperation, the game-layer network, the strategy-diffusion layer, and the payoff matrix design. We found that the optimal incentive is node-wise, unique to each node on the network, influenced by its relative position according to the adjacency matrices of both layers. We also found that the optimal solution is not exclusively reward or punishment. While one node can receive a reward, the other node may receive punishment at the same time, depending on its current level of cooperation, relative to other nodes. We provide analytic solutions and numerical validations for the cases studied, comparing our results to the existing literature.

Many operational decisions involve alternatives with several uncertain attributes. When comparing such alternatives, stochastic dominance requires every decision maker in a prescribed class to prefer one alternative to the other. These orders have two known limitations: a single extreme decision maker can rule out dominance, and multivariate dominance can be difficult to verify. To address these limitations, we develop two relaxations of their standard representations: compensated stochastic dominance (CSD) relaxes Strassen's coupling representation, while reference-weighted stochastic dominance (RWSD) relaxes the integral representation. We characterize the utility class generated by CSD, recovering known almost stochastic dominance orders as special cases, and characterize it for RWSD in several cases. Importantly, we show that CSD and RWSD can each be checked by computing a single value, an optimal-transport value for CSD and a CVaR value for RWSD, and verifying that this value is nonpositive. This inequality characterization allows us to derive finite-sample guarantees for both orders, even when the outcome dimension is large. An application to Olist e-commerce data shows that, even when empirical FOSD fails, CSD and RWSD quantify how small this failure is and what is needed to overcome it.

We prove that the $\mathrm{iSWAP}$ gate maximises the spectral gap of the Hermitian second-moment operator on every connected graph with at least three vertices, among all two-local unitary circuit ensembles. We further prove that the polyhedral cone defined by asymmetric four-point inequalities is invariant under the transpose of the $\mathrm{iSWAP}$ semigroup, yielding a componentwise comparison certificate for its positive Perron–Frobenius eigenvector. These results resolve a conjecture of Kong, Li, and Liu.

We study generalized backward stochastic differential equations (GBSDEs) and generalized reflected backward stochastic differential equations (GRBSDEs) on a general filtered probability space satisfying the usual conditions, without assuming that the underlying filtration is quasi-left-continuous. The equations are driven by a prescribed predictable, bounded, nondecreasing RCLL process \(A\), which acts as a possibly discontinuous stochastic clock, which we call a driver. We first establish a priori estimates, stability, existence, and uniqueness results for GBSDEs whose generator is Lipschitz continuous with respect to the state variable. Since \(A\) may have jumps, the analysis is carried out in weighted spaces defined through the stochastic exponential \(\mathcal{E}(\beta A)\). We then investigate GRBSDEs with an optional regulated lower obstacle. When the generator is independent of the state variable, we develop two complementary approaches. The first relies on a Snell-envelope representation and optimal stopping arguments, while the second is based on a modified penalization procedure adapted to the discontinuities of the right jumps of the obstacle. The general Lipschitz case is subsequently obtained through a fixed-point argument in an appropriate weighted Banach space.

For a class of weighted degenerate or singular problems on a cylinder, Almgren-type monotonicity methods are employed to derive asymptotic estimates of solutions near the edge between the base and the lateral surface, where a Robin boundary condition is imposed. As a relevant application, local asymptotics and unique continuation from the boundary are obtained for the Robin and Neumann spectral fractional Laplacians. The local decay rates of the solutions are the same for both the Neumann and the Robin problems and are explicitly quantized by a weighted spherical spectral problem with some symmetry.

This paper presents a robust tube model predictive control (MPC) framework for nonlinear systems represented by bilinear Koopman models identified from data. We derive discrete-time robust control contraction metrics (RCCMs) for bilinear Koopman models, which certify a contraction property that explicitly accounts for the mismatch between the Koopman model and the true dynamics. To synthesize such certificates for the typically high-dimensional lifted state, we propose a scalable learning approach that trains neural network parameterizations of the metric and of an associated feedback controller over a sparsely sampled subset of the lifted state space. The resulting certificates are used to construct a robust tube MPC scheme with tightened constraints, for which we establish recursive feasibility, robust constraint satisfaction and input-to-state stability (ISS) of the closed-loop system with respect to the mismatch between the Koopman model and the true dynamics. By exploiting the linear parameter-varying structure of bilinear Koopman models, the MPC problem becomes convex for a fixed scheduling sequence, while the geodesic computation underlying the contractive feedback is reformulated, via a Chebyshev pseudospectral discretization, as a sequence of quadratic programs. The complete framework is validated on a nonlinear pendulum benchmark with a state-dependent input gain.

We study multi-hop visibility inside the vacant set of three obstacle models in $\mathbb R^d$ with slow decay of spatial correlations and disparate obstacle geometries: Poisson-Boolean models with general i.i.d. radii distributions, Poisson cylinders and Brownian interlacements. For any $N\geq 0$, we obtain sharp bounds on the probability $P_{\mathrm{vis}}^N(r)$ of visibility to distance $r$ via $(N+1)$ hops in terms of the (explicit) probability of direct visibility to distance $r$ in a given direction, generalizing our earlier result from arXiv:2304.10298 for $N=0$. We observe a universal behavior of $P_{\mathrm{vis}}^N(r)$ in terms of two characteristic scales. In the three models of interest, these scales are generally the same, but with some anomalous exceptions in low dimensions.

Let $h$ be a positive integer, and let $A$ be a finite set of integers. We derive an exact formula for $|hA|$. Furthermore, let $A=\{0,1,\ldots,s,a,b\}$, $1\leq s<a<b$, and write $b=qa+r$ with $0\leq r<a$. By using generating function, we prove that $|hA|$ equals a definite explicit formula expressed in terms of certain truncated binomial coefficients for all positive integers $h$ if and only if $r=0$ or $qs+r\geq a$. This generalizes a result of Nathanson.

We study signature transformations of heterogeneous paths $Y=(A,X)$ whose components may differ in regularity and probabilistic structure. We introduce an invertible change of coordinates $\Psi$ such that the transformed signature $\Psi\circ\mathrm{Sig}$ eliminates mixed integration against the irregular component $X$ and admits a representation in terms of signature coordinates of $X$ and iterated integration against the regular component $A$. In addition, we exploit this representation to further represent partially symmetrized signatures. Our main application concerns new expected signature formulas for processes with deterministic augmentation. On the analytical side, these formulas are leveraged to study moment problems. On the numerical side, they enable accurate computation of expected signatures, thereby overcoming typical computational bottlenecks in applications. We illustrate these advantages in a signature-based stochastic control problem driven by fractional Brownian motion.

Because 2SLS is built from sample averages, a small number of observations can have a disproportionate effect on estimates and inference. We introduce W-2SLS, a simple drop-in robustification that replaces these averages by quantile-winsorized means. We analyze W-2SLS under adversarial contamination, which permits both the identities and the reported values of the contaminated observations to depend on the realized clean sample and therefore accommodates targeted or strategic manipulation. Under finite $m$-th moments, W-2SLS attains the minimax-sharp rate $\eta_{n}^{1-\frac1m}+n^{-1/2}$, where $\eta_n$ is the fraction of observations that may be altered. Matching lower bounds identify the exact contamination thresholds for uniform consistency, root-$n$ estimation, and centered Gaussian inference with the same first-order law as clean-sample 2SLS. When $\sqrt{n}\eta_{n}^{1-\frac1m}\to 0$ robustness is first-order free. We also construct feasible heteroskedasticity-robust inference and a winsorized Anderson–Rubin test valid under weak identification and adversarial contamination. Finally, even without contamination, ordinary 2SLS can have poor uniform finite-sample concentration, whereas W-2SLS admits confidence-calibrated sub-Gaussian deviation guarantees.

We study a Nelson-type Hamiltonian describing a massive spinless non-relativistic particle moving in a planar soft quantum waveguide and linearly coupled to a massive scalar bosonic field. Our spectral analysis relies on comparing the curved waveguide with its straight counterpart. For the straight Hamiltonian, longitudinal translation invariance yields a fiber decomposition, and we prove that its spectrum is purely essential and gapless. For the curved Hamiltonian, we establish an HVZ-type formula for the bottom of the essential spectrum, involving that of the straight system and the one-boson threshold. We also derive a sufficient condition for the existence of non-empty discrete spectrum in the curved case, expressed in terms of an effective one-dimensional Schrödinger operator.

We investigate the forward-backward-splitting version of the Plug and Play (PnP) method for linear ill-posed problems with MMSE estimators as denoisers. In contrast to existing literature, we consider estimators which are specialized for (degenerate) Gaussian noise with possibly non-diagonal covariance matrices. We further deviate from the classical iteration by replacing parts of the descent step with a linear operator that relates the observation noise to that of the MMSE estimator. Under mild assumptions, we derive several properties of the denoiser and prove recovery guarantees of the iteration both pointwise and in the Wasserstein distance of the underlying probability distributions. Crucially, our analysis shows that the denoiser cannot be chosen in a physics-agnostic way, that is, independently of the forward model. We extend our results to the case where the MMSE denoiser is parametrized by a neural network and derive the corresponding recovery bounds.

We show that for any $n \geq 1$ and any convex body $K \subset \mathbf{R}^n$, there exists $T \in \mathrm{SL}(n)$ such that \[ \mathrm{diam}(TK) \lesssim \sqrt{n} \mathrm{vr}(K), \quad \mbox{and} \quad M^\ast\Big(T(K-x) \cap r\mathrm{vr}(K)\,B^n_2\Big)\lesssim \mathrm{vr}(K) \sqrt{\log(\mathrm{e} r^2)}, \] for every $x \in K$ and every $r \geq 1$. Above, $M^\ast(\cdot)$ denotes the spherical mean width and $\mathrm{vr}(\cdot)$ denotes the volume radius. As a consequence, we establish for any convex body $K \subset \mathbf{R}^n$ that \[ 1 \leq \inf_{T \in \mathrm{SL}(n)} \, \frac{M^\ast(TK)}{\mathrm{vr}(K)} \lesssim \sqrt{\log(\mathrm{e} n)}. \] The estimates above are sharp, up to universal constants, as they are attained for the crosspolytope and any regular $n$-simplex. Up to universal constants, our results show that all quermassintegrals, and the logarithm of the covering numbers for all scales, simultaneously for $K$ and the polar body $K^\circ$, are maximized by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization. To establish the results, we work with an extension of Bobkov's maximal Gaussian measure position to possibly non-symmetric convex bodies. Our results imply that this position is an optimal "regular" Milman position, thereby improving a result of G. Pisier. We establish a non-symmetric analogue of the strong Gaussian (B)-theorem which may be of independent interest.

Mapping Calabi-Yau geometry to early Universe cosmology remains a primary goal in string phenomenology. We present an inflationary mechanism where flux stationarity shifts the field dynamics, driving inflation along the volume-controlling saxion direction near a Hodge boundary. In this region, the internal geometry strictly dictates the scalar potential. This setup stabilizes the field against heavy moduli corrections without consuming extra flux tadpoles. As a result, the properties of the geometric boundary translate directly into primordial tensor perturbations, creating a clear physical link from string compactifications to measurable data. The resulting geometric predictions match current cosmic microwave background limits and provide specific, testable targets for upcoming LiteBIRD and CMB-S4 observations.

We provide a framework for robust mean field control problems that describe multi-dimensional optimal liquidation problems under uncertainty from both the underlying stochastic process and the deterministic model parameters. The verification results are established with Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations where the variables are probability measures and the Hamiltonian nonlinearly involves the joint distribution of position and momentum. Using novel a priori estimates, we establish the well-posedness of the HJBI equations featuring general or quadratic Hamiltonians that are neither displacement convex nor concave in their momentum. The a priori estimates and well-posedness results are extended during their application to optimal liquidation problems, where we allow the Hamiltonian to have derivatives of linear growth and solve the constrained multi-dimensional linear quadratic optimal liquidation problem under composite uncertainty.

The Berge-Fulkerson Conjecture states that every bridgeless cubic graph has six perfect matchings such that each edge belongs to exactly two of them. This conjecture has remained open since 1971, and several of its weakenings have been proposed over the years. Two of the most prominent are the Fan-Raspaud Conjecture and the Máčajová-Škoviera Conjecture. It is well known that the Berge-Fulkerson Conjecture implies the Fan-Raspaud Conjecture, which in turn implies the Máčajová-Škoviera Conjecture. These problems have been studied for years in the context of finite graphs, and many equivalences between them and other results have been established. However, little to nothing is known about them in the context of infinite graphs. In this paper, we investigate whether these conjectures remain valid in the infinite setting, establish their implications in analogy to the finite case, and prove that their finite versions are equivalent to their respective infinite versions.

Robust multiple testing procedures for assessing equality restrictions on the coordinates of high-dimensional mean vectors are proposed. Our procedures are based on quantile-winsorization techniques, approximately control the familywise error rate (strongly) under adversarial contamination, and allow the number of hypotheses to grow exponentially with sample size, despite requiring only slightly more than two moments. Technically, our one-sample results build on recent Gaussian approximation inequalities for the distribution of high-dimensional quantile-winsorized means, whereas our two-sample results are based on extensions thereof, which we develop here and which could be of some interest in their own right.

We prove the Alessandrini-Bassanelli conjecture on nilmanifolds with nilpotent complex structures and on holomorphically parallelizable solvmanifolds. As a consequence, we classify holomorphically parallelizable solvmanifolds admitting $p$-Kähler structures, for lower values of $p$. We further provide new examples of $(n-2)$-Kähler manifolds, in the setting of compact holomorphically parallelizable manifolds with reductive universal cover.

We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in $\bf{R}^4$, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We describe three regimes with sharp borderlines. In narrow cones the volume grows polynomially with exponent $n+O(\varepsilon^2)$, and the order $\varepsilon^2$ is attained; in wide cones the growth can be arbitrarily fast. For holomorphic curves in $\mathbb{C}^2$, there are also three regimes and the borderline is determined by the Kähler angle.

Global-local priors, often also referred to as shrinkage priors, have proved to be a very effective tool for the analysis of high dimensional data under sparsity. Asymptotic theoretical properties of such priors, studied under various scenarios, are now available in the literature. However, to our knowledge, theoretical guarantees of such priors provided so far, involve the assumption of known sample variance. The present paper relaxes this assumption, and carries out the analysis with a prior assigned to the error variance as well. In the process, some new tail bounds for shrinkage factors are developed, and these results are then utilized in providing asymptotic minimax rates for the posterior means of the parameters of interest.

The $q$-state ferromagnetic Potts model on a graph $G$ is a probability distribution on all $q$-colourings of $G$ that favours many monochromatic edges. Approximate sampling from the Potts model is a central problem in the study of spin systems on sparse graphs, especially in the low-temperature regime, where the model strongly favours ordered configurations, often creating bottlenecks that make Markov-chain sampling inefficient or difficult to analyse. We focus on the sparse random graph $G(n,d/n)$. The local neighbourhoods of $G(n,d/n)$ are tree-like, but the relevant underlying graph is a Poisson Galton-Watson tree. This motivates the study of Glauber dynamics for the low-temperature Potts model on such trees with monochromatic boundary conditions. The Poisson setting introduces difficulties absent from the regular case: degrees fluctuate, long induced paths may appear, and branches can terminate before reaching the boundary. As a result, the effect of the monochromatic boundary at the leaves is much less uniform. Our main result shows near-linear mixing for the Glauber dynamics on Poisson trees with monochromatic boundary conditions. This extends the corresponding regular-tree results of Martinelli, Sinclair, and Weitz (SODA 2004) and of Blanca, Chen, Stefankovič, and Vigoda (RANDOM 2021) to the irregular trees arising from sparse random graphs. Our proof introduces an adaptive block decomposition of the tree, built around regions containing large regular subtrees, and combines it with correlation-decay estimates and functional-inequality arguments. We also obtain a near-linear-time approximate sampling algorithm for the Potts model on $G(n,d/n)$ at all temperatures, speeding up the best previous algorithm of Galanis, Goldberg, and Smolarova (ICALP 2025). The main new ingredient is a refined analysis of the low-temperature regime, building on the Poisson tree result.