On Zeros and Algorithms for Disordered Systems II: The Antiferromagnetic Ising Model up to Kesten-Stigum
Ferenc Bencs, Brice Huang, Daniel Z. Lee, Kuikui Liu, Guus Regts
AI 速览：作者提出首个在均匀随机 d-正则图上、对反铁磁 Ising 模型在直至 Kesten-Stigum 阈值的全部逆温度范围内以任意高精度估计配分函数的多项式时间算法。
We give the first polynomial-time algorithm to estimate the partition function of the antiferromagnetic Ising model on the uniformly random $d$-regular graph to arbitrarily high accuracy for all inverse temperatures up to the Kesten-Stigum threshold. This is the entire regime in which the model is "replica symmetric," and is conjectured to be optimal for polynomial-time counting and sampling. To achieve our algorithmic result, we establish the absence of Fisher zeros for the partition function in a constant-width strip around the interval from $0$ up to the Kesten-Stigum threshold, with high probability over the randomness of the input graph. AI Disclosure: The overarching vision and proof architecture were developed solely by the authors without AI input. GPT 5.5 Pro was used, with extensive guidance from the authors, for only a few minor technical aspects of the proof. The authors independently verified correctness and wrote everything by hand. We take full responsibility for its contents. A more detailed statement can be found in Section 1.2.
Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics
Francesco d'Amore, Joel Rybicki, Nobutaka Shimizu, Takeharu Shiraga, Robin Vacus
AI 速览：该论文研究同步 h-Majority 共识动力学中样本数 h 与达成共识所需时间之间的权衡，并给出近乎最优的上界与匹配下界。
How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-\textsf{Majority}, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-\textsf{Majority} achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $\widetilde{O}\left(1+\min\left\{\frac{1}{\min\{h^2p_{\max},1\}},\frac{\sqrt{n}}{h}\right\}\right)$ rounds, where $n$ is the number of nodes and $p_{\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-\textsf{Majority} that holds for every $h\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\log n/\min\{h^2p_{\max},1\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.
Advanced power adjusting block withholding attacks such as PAW and T-PAW have been analyzed so far under a global-reset version of the proportional scheme (GRP), which is rarely used in practice. In this paper, we analyze T-PAW, which contains FAW and PAW as special cases, under the block-dependent payout schemes deployed in practice: Pay-Per-Last-N-Shares (PPLNS), round-based Proportional (PROP) and exponential-decay (EXP). We derive the revenue ratios analytically, by renewal arguments for PPLNS and through the stationary carry process of an open round for PROP and EXP. We further introduce Power Adjusting Hopping (PAH), where an adversary of any size adjusts its power allocation towards the pool according to the blockwork age of the round without withholding any block, and derive its revenue ratios under PROP and EXP. We show that GRP misestimates the T-PAW gain in a scheme-dependent direction: it overstates the gain under PROP up to an order of magnitude and understates it under EXP for the original slush decay rate. PROP is highly vulnerable to PAH, while EXP faces a trade-off: a slow decay invites PAH and a fast decay invites T-PAW. PPLNS is the least vulnerable scheme: it is immune to PAH and it mitigates, but does not eliminate, T-PAW. All analytical results are verified by Monte Carlo simulations.
Zero-freeness for Random Regular Graphs up to the Replica Symmetry Breaking Threshold
Zejia Chen, Chunyang Wang, Xinyuan Zhang
AI 速览：作者在随机 d-正则图上证明硬核模型与反铁磁 Potts 模型（含正常 q-染色）在副本对称区内具有零点自由区域，并由此得到多项式时间近似配分函数的算法。
We study antiferromagnetic spin systems on random regular graphs through the lens of the phase transitions characterized by the complex zeros of the partition function. Our motivating examples include the hardcore model and the antiferromagnetic Potts model (including proper $q$-colorings at $β=0$). For both models on random $d$-regular graphs, we establish, with high probability over the graph, zero-free regions throughout the interior of the replica-symmetric regime. Asymptotically, this regime extends up to tree occupation density $$α< α_{\mathrm{RSB}}(d) = (2\log d - 2\log\log d + 2 - 2\log 2 + o_d(1))/d$$ for the hardcore model and up to degree $$d < d_{\mathrm{RSB}}(q) = (2q-1)\log q - 2\log 2 + o_q(1)$$ for proper $q$-colorings. Leveraging the standard algorithmic implications of zero-freeness, we obtain deterministic polynomial-time algorithms that multiplicatively approximate the corresponding partition functions at fixed rational parameters throughout this regime, with high probability over the graph. Our algorithmic guarantees significantly improve upon prior work and extend beyond the parameter ranges reached by previously known search algorithms. To establish our results, we first develop reinforced small subgraph conditioning, which resolves short-cycle fluctuations in the complex plane by lifting classical small subgraph conditioning on random regular graphs to random holomorphic functions. Second, we overcome the standard second-moment barrier caused by the exponential blowup of the bare second moment inside the replica-symmetric phase, by showing that replica symmetry implies a nonpositive Bethe functional and upgrading planted overlap concentration to an exponentially small bound, which enables a typical-graph truncation that controls complex second moments up to the replica symmetry breaking threshold.
Sampling Sphere Packings with Continuum Glauber Dynamics
Aiya Kuchukova, Santosh S. Vempala, Daniel J. Zhang
AI 速览：作者为排斥对势的Gibbs点过程建立连续Glauber动力学的谱隙，并据此给出任意程对势的高效采样算法，改进了固定尺寸/密度堆积的可采样阈值。
Continuum Glauber dynamics is a spatial birth-death process whose stationary distribution is a Gibbs distribution. We establish a spectral gap for Continuum Glauber dynamics applied to Gibbs point processes with repulsive pair potentials, a well-known special case of which is the hard sphere model. For arbitrary-range repulsive pair potentials, we show that a continuous version of Spectral Independence suffices to establish a spectral gap. This extends the regime of activity for which Continuum Glauber dynamics is known to mix, yielding a simple efficient sampling algorithm for arbitrary-range pair potentials that matches the known efficient sampling regime for finite-range pair potentials currently based on specialized algorithms. As a consequence, we also improve the threshold up to which packings of fixed size/density can be efficiently sampled from a bounded domain, the first improvement since Kannan, Mahoney and Montenegro (2003). To prove these results, we develop continuous analogs of Spectral Independence and negative fields localization. We show that a stronger variant of zero-freeness implies Spectral Independence, which in turn allows us to run the localization scheme to boost the spectral gap of Continuum Glauber dynamics from smaller activity to larger activity. While this follows the high-level blueprint of Chen and Eldan (2022) for the discrete setting, we have to address several novel difficulties due to the continuous setting. Notably, we avoid discretization in the algorithm and the analysis and work directly in the continuous setting.
A robot searching for the source of an odor must eventually answer a notably hard question: have I arrived? Near the source, concentration gradients homogenize, turbulence corrupts the signal, and a small robot often lacks the positional accuracy to map the plume. Here, we describe a lightweight remedy inspired by a World War II statistics conundrum: the German tank problem. To approximate German armament production, Allied analysts developed an estimator based on serial numbers from captured tanks. Analogously, we use the same procedure over sampled chemical sensor readings from a robot to create estimates for the peak concentration of an odor plume, and therefore the odor's source. Treating each sensor reading as an observation, our unbiased estimator needs only a running maximum and a sample counter. A confidence bound turns both numbers into a principled rule that the robot or UAV can use to determine whether it should keep searching for the odor source. We derive the estimator and its bound, demonstrate its sample efficiency, and examine its robustness even when its assumptions fail. Finally, we summarize the estimator's role as the termination criterion for olfactory navigation.
We develop a unified framework for the instantaneous total level crossing rate (LCR) of products of independent positive nonstationary fading processes. Universal upper and lower bounds are first derived from an exact decomposition of the product derivative and signed crossing imbalance, requiring only factor-wise crossing statistics. We then introduce the drifting Gaussian-derivative (DGD) class, in which each conditional derivative is Gaussian with possibly nonzero drift. For products of DGD factors, an exact Gaussian-mixture representation yields bounds that jointly capture aggregate drift and derivative fluctuations. The lower bound reduces under stationarity to a known $\ell_2$ bound, whereas, to the best of the author's knowledge, the upper bound is new even in the stationary case. Furthermore, the bounds are asymptotically exact in drift-dominated regimes. The framework is specialized to nonstationary Double-Rayleigh fading and to quasi-identically distributed DGD factors, with closed forms for products of $α$-$μ$ envelopes. Numerical results show that the DGD bounds closely bracket the exact LCR.
Accelerated dynamics methods, such as parallel replica dynamics, require estimates of the decorrelation time, after which a process trapped in one of its metastable states is distributed according to the quasi-stationary distribution (QSD). In a method proposed by Binder, Lelièvre and Simpson, decorrelation times are defined by coupling a particle approximation of the non-linear conditioned probability flow of the process with MCMC-inspired diagnostics. We explore an alternative to MCMC diagnostics, based on coarse-graining of non-linear probability flows and simulation-based inference. We first develop a coarse-graining framework for killed Markov jump processes conditioned on survival, proving uniform-in-time, non-asymptotic estimates approximating the non-linear probability flow of the true process by the non-linear flow of a surrogate process, defined with a collective variable. The framework does not assume reversibility of the true process, and also gives bounds which scale explicitly in an adiabatic regime. Motivated by these results, we then construct a quasi-stationarity diagnostic using simulation-based inference: a recurrent neural network (RNN), trained on particle approximations of non-linear effective flows, estimates posterior probabilities of decorrelation. The learned diagnostic is then transferred to collective-variable traces of high-dimensional particle approximations. We investigate the behavior and robustness of this diagnostic on a set of numerical experiments, comparing it to the MCMC diagnostic of Binder et al.\ on both synthetic examples and a molecular system, in and out of its training prior distribution. When its prior assumptions hold, the RNN diagnostic outperforms the MCMC diagnostic in ROC analysis, is less conservative, fails less often, and has an interpretable decision threshold.
The universal approximation property uniformly with respect to weakly compact families of measures is established for several classes of neural networks. To that end, we prove that these neural networks are dense in Orlicz spaces, thereby extending classical universal approximation theorems even beyond the traditional $L^p$-setting. The covered classes of neural networks include widely used architectures like feedforward neural networks with non-polynomial activation functions, deep narrow networks with ReLU activation functions and functional input neural networks.
We show that the transformer architecture emerges as the score of last-token laws in a natural statistical model of language, where sentences are empirical measures of tokens, a language is a law on $P(S^d)$, and our prior belief is that sentences carry no structure at all. By the maximum-entropy principle, we select the law closest to this prior in relative entropy among those reproducing an observed sentence-level statistic. We discuss the relevance of the statistic in the posterior interaction between tokens and we build a statistic, able to capture that interaction, with Luce's choice axiom truncated at the lowest relational order. This procedure gives the softmax choice rule, with the attention cross-moment as its mean parameter. The query--key matrix is then the Lagrange multiplier of a maximum-entropy problem inside each sentence, and the value matrix the multiplier of the constraint over the language, so that the two are untied. Under the optimal entropic law we then show that a transformer head appears in the score of the last-token law. For ordered sentences, causal masking follows from the masked reference choice rule, and with a next-token cross-moment the score equals one causally masked attention head. We also discuss multi-head attention, weight-tied unembedding, and a heuristic reading of residual blocks as projected Langevin steps.
Pekka Malo, Lauri Viitasaari, Antti Suominen, Eeva Vilkkumaa, Olli Tahvonen
In many control problems, actions consume a resource budget, such as a wear allowance, and the probability of exhausting it within a given horizon must stay below a tolerance. A simple, widely used approach is to augment the state with the remaining budget, subtract an interior barrier from the utility and penalize exhaustion, so that policy-gradient methods can be applied unchanged. We clip the combined one-step reward from below at a level m to keep the returns of exploratory policies finite. The clip level is a price of failure risk, since the clipped value decomposes exactly into the utility before failure, the cost of the interior margin and m times a geometric failure risk. Consequently, an approximately optimal policy maximizes utility net of margin cost among policies with no greater risk. Its risk exceeds that of any comparison policy by at most order 1/m and can be certified from its value or from validation data. Our main result proves exponential convergence of the population policy-gradient flow for every parameter regularizer that is convex along generalized geodesics and sufficiently strong relative to the price. This extends the mean-field analysis from relative entropy to weight decay and to finite networks. Relative entropy adds parameter noise and a pull toward a prior, whereas weight decay provides only the pull. Action regularization bounds changes in failure risk by the relative entropy to a reference policy. The bounds are evaluated on an exact maintenance benchmark, a fuel-tank controller, a solvable flow and a benchmark-suite task.
Triadic closure is a natural mechanism for modeling the clustering phenomenon observed in many real-world networks. Given a possibly random undirected graph $H$, we consider pairs of non-adjacent vertices that have a common neighbor. Conditional on $H$, each such pair is joined by an edge independently and simultaneously, with a probability depending on the degrees of its endpoints and on the number of their common neighbors. The resulting graph is called a triadic closure of $H$. We study the effect of this operation on local convergence. More precisely, we prove that if a sequence of potentially random finite undirected graphs of $\left( H_n \right)_{n \in \mathbb{N}}$ converges locally in probability to a rooted graph $H^*$, then the corresponding sequence of triadic closures converges locally in probability to the same triadic closure of $H^*$. As a consequence, we obtain a general limit theorem for the clustering coefficients of triadic closures. In particular, we show that the triadic closures can have a strictly positive limiting clustering coefficients even when the limiting clustering coefficients of the parent graph sequence is zero. Finally, we apply our results to several classical sparse random graph models including Erdős-Rényi, configuration, preferential attachment models and compute the limiting clustering coefficients of their triadic closures.
Inspired by birds flying through cluttered environments, Karaman and Frazzoli showed that high-speed flight through an ergodic random forest exhibits an ultimate speed limit: above a critical speed, almost surely no infinite collision-free trajectory exists for a single bird. This paper formulates a corresponding bulk-transport problem for many birds. We construct an explicit directed corridor network in a Poisson disk forest and impose a common temporal headway at its waypoints. The maximum current equals a minimum vertex-cut capacity and is achieved by repeated flight along vertex-disjoint paths. The same schedule maximizes completed traversals at every finite horizon, with constant forward speed and no internal waiting. For a forest belt of fixed depth, exact transverse superadditivity yields an almost-sure deterministic current-density limit. A uniform shape theorem for an asymmetric planar dual identifies the current when depth and width grow together. For diverging depth bounded by a constant times width, this current equals the width-first, depth-second limit and is independent of the positive aspect ratio. When diagonal corridors are available, bulk current is positive at sufficiently low obstacle intensity and zero at sufficiently high intensity. For stochastic demand, an upstream admission queue is stable at every rate below the finite-network capacity.
We study Gibbs averaging of finite Markov chains with a prescribed stationary distribution $π$. Given a transition matrix $P$ and the Gibbs kernel $G$ induced by a state-space partition, we prove that $GPG$ is an information projection for every admissible Bregman generator $φ$, with divergences defined using transition probabilities relative to $π$. A Pythagorean identity characterizes this projection and recovers the relative entropy and weighted Frobenius cases. A dual characterization identifies $GPG$ as a kernel closest to stationarity among those with the same projection chain as $P$; for relative entropy, it maximizes the entropy rate. The identities also give monotonicity under partition coarsening. For every reversible ergodic baseline $P$, Gibbs averaging cannot increase the squared weighted Frobenius distance to stationarity at any positive integer time, and the inequality is strict unless $P=GPG$. The corresponding mixing time therefore cannot increase. To choose the partition, we identify the divergence of $G$ from stationarity as the exact worst-case divergence of $GPG$ over all $π$-stationary baselines $P$. This objective depends only on the block masses. We give an exact algorithm for concave block costs, including those arising from relative entropy and power generators $φ(u)=u^p$ with $1<p<2$, and a greedy approximation algorithm with a provable guarantee for $p>2$. For power generators $φ(u)=u^p$ with $p>1$, the objective is proportional to Tsallis entropy of order $2-p$: its minimization favors imbalanced block masses when $1<p<2$ and balanced block masses when $p>2$, while all $k$-block partitions have the same objective value when $p=2$. A numerical experiment compares the resulting projected chains with a local Metropolis sampler on a multi-modal $π$.
Particle Gibbs samplers with backward sampling are widely used for sampling the smoothing distribution of state-space models, but, with few particles, the new trajectory often coincides with the current one at many time steps. A known but rarely used remedy replaces the random selection of the final particle by a Metropolis--Hastings move which avoids the current particle. We show that, with backward sampling, this remedy only helps at the last time steps, because its effect on the update rates decays geometrically with the distance to the final time. We propose instead Metropolized backward sampling, which makes such a move at every step of the backward pass, forcibly switching trajectories whenever possible. Additionally, we introduce a coupled version which prevents proposing the current trajectory, maximizing the chances of an actual switch versus the current trajectory and performing perfect sampling when the state-space model has no observations, a feature not shared by backward sampling. We prove that, at every time step, both methods dominate backward sampling as they update the state more often and have smaller lag-one autocorrelations than it does. Our methods are drop-in replacements for backward sampling, and can be used to improve any particle Gibbs sampler.
Let $X$ be a random vector in $\mathbb{F}_2^m$, and let $X'$ be an independent copy. We prove that, for every $\varepsilon>0$, there is a subspace $G$ with $\dim G\le 2^{320(1+\varepsilon)/\varepsilon}H(X)$ such that $H(q_GX)\le(1+\varepsilon)(H(q_GX+q_GX')-H(q_GX))$. In particular, the quotient entropy is at most $(1+\varepsilon)(H(X+X')-H(X))$. The coefficient is independent of the ambient dimension, and its exponential order in $1/\varepsilon$ is necessary. The structural result is a logarithmic rigidity estimate for finite weighted families of laws with averaged subspace entropy sparsity. A common variational quotient controls both a family and its convolution family without decreasing the original family's normalized mean entropy defect. Posterior families preserve averaged sparsity, and a fresh-index argument combines a local-to-global estimate with the entropic Marton theorem to increase the defect ratio at a fixed multiplicative sparsity cost. We also obtain two-input almost-additivity and concentration of moderate-doubling sets on subspace cosets at the same exponential scale. The constants are not optimized.
Let $P$ be an irreducible reversible Markov kernel on an $m$-state space $Ω$, and denote its right spectral gap by $\Gap(P) =1-λ_2(P)$. For a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\widetilde P_n$. For each fixed $m\ge2$, define \[ c_m^\star= \inf_{\substack{|Ω|=m,\ P\ \mathrm{irreducible\ and\ reversible} n\ge2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \] We prove \[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \] We also show $q_{m-2}=(m+\log m+O(1))^{-1}$, which implies that $c_m^\star=Θ(m^{-1})$. Thus, the reciprocal optimal comparison coefficient $C_m^{\mathrm{opt}}=(c_m^\star)^{-1}$ is of the order $m$. In the two-state case, we determine the exact values $c_2^\star=1/4$ and $C_2^{\mathrm{opt}}=4$. %the exact constants for $m\ge3$ remain open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\Gap(\widetilde P_n)=Θ_P(n^{-1})$.
Redundant context affects attention in two distinct ways: multiplicity adds normalization mass, while score variation within correlated groups can produce extreme individual scores. We separate these effects in a planted retrieval model with exponentially many Gaussian distractors organized into supplied clusters whose number and size both grow exponentially with dimension. The model is a two-level generalized random energy model, whose classical free energy yields exact retrieval thresholds. Gaussian interpolation further shows that increasing positive correlation at fixed token count lowers the distractor free energy, unlike adding duplicates. Our main result establishes a sharp limitation of count correction, which subtracts the log cluster size from pre-softmax scores: this correction removes multiplicity but retains within-cluster extremes. Even when the inverse temperature is chosen after observing all scores, count correction requires a signal at least $\sqrt{2α_1}$, where $α_1$ is the exponential growth rate of the number of clusters. By contrast, averaging scores before exponentiation suppresses token-level fluctuations and lowers the optimal signal threshold to $\sqrt{2ρα_1}$, where $ρ$ is the within-cluster correlation. This gap persists after adding exponentially many singleton distractors, so group size alone does not identify the target, and extends, under explicit cumulant regularity assumptions, to fixed finite hierarchies of independent-increment scores. Finite-sample, stability, and participation results complement the asymptotic analysis. A controlled retrieval experiment with frozen sentence embeddings illustrates both sides of the distinction: averaging helps when the relevant item is a singleton, but can discard useful evidence when a relevant group contains multiple matching items.
Allowing for small fraction of errors in retrieved stored binary patterns, Hopfield-Hebbian neural network (NN) associative memory dynamics [14] achieves linearly scaled capacity. The famous proportionality constant $α_c\approx 0.14$ is obtained in a large dimensional scenario as the ratio of the number of reliably stored patterns $m$ and the patterns' length $n$ under the assumption that uniformly randomly chosen binary patterns are \emph{independent} among themselves. While such an idealized scenario is now well understood, more practical scenarios with correlated/noisy datasets remaining challenging even for classical statistical mechanics replica approaches. We consider precisely such practical settings where each \emph{archetype} pattern is replaced with $M$ correlated \emph{examples} with parameter $r$ governing the quality of correlation. We develop a powerful Random Duality Theory (RDT) based framework to analyze correlated data and already on the first level of RDT lifting obtain capacity characterizations that remarkably well match behavior exhibited through simulated dynamics. Moreover, characterizations explicitly feature correlation parameters $M$ and $r$, which allows to precisely determine their effects on phase-transitioning behavior of the underlying dynamics. Our results are also a well-fitting complement to the existing literature. Namely, to circumvent challenging statistical mechanics analyses, an alternative combination of spectral and replica methods is considered in [4]. We confirm that [4]'s elegant spectral characterizations are excellent approximations in the low correlated regime. At the same time, we also close the gap in the regimes with higher correlation levels where spectral methods deviate from the practical dynamics performance.
Every semi-magic square of order $n$ satisfies the axial moment identities $\sum_{i,j}s_{ij}j^m=\sum_{i,j}s_{ij}i^m=M(n)\sum_jj^m$ for all $m\ge1$, where $M(n)=n(n^2+1)/2$. For real matrices the converse requires $1\le m\le n-1$, whereas for arrangements of $\{1,\dots,n^2\}$ fewer moments suffice once $n\ge14$. We define the permutation threshold $k^*(n)$ as the least $k$ such that every arrangement satisfying these identities for $1\le m\le k$ is semi-magic. In terms of the column-sum deviations $γ_j$, the degree-$k$ conditions state that $\sum_jγ_jt^j$ is divisible by $(t-1)^{k+1}$, linking $k^*(n)$ to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives $k^*(n)\le\sqrt{2(n-1)\log(n^4/4)}$, we prove the converse realisability bound $k^*(n)\ge(\sqrt2-o(1))\sqrt n$ (and $k^*(n)\ge(\sqrt6-o(1))\sqrt n$ when $4\mid n$) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that $k^*(n)=n-1$ for $3\le n\le13$, $k^*(n)=n-2$ for $14\le n\le17$, and $k^*(n)\le n-3$ for $n\ge18$, and determine $k^*(n)$ for $18\le n\le40$ with computer assistance. Finally, at the $k=1$ level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy $E_{\mathrm{full}}$, for the four-term energy $E_4$ combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that $|L_4|=14{,}639{,}440$ arrangements of order $4$ satisfy $E_4=0$.
We consider the tensor principal components analysis (tensor PCA) problem on the sphere, in which we are asked to recover an unknown vector in $\mathcal{S}_N=\{x\in\mathbb{R}^N:\lVert x\rVert^2=N\}$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. Continuing the program initiated in Part I [arXiv:2610.16705], we show that spherical Langevin dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization. Together with Part I, our results demonstrate that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.
We study conformal prediction under covariate shift with a known covariate density ratio, seeking prediction sets with minimum expected length under the target distribution. For every calibration sample size, we prove that the rule combining the inverse conditional-density score with randomized weighted conformal calibration minimizes this expected length among all rules valid over the full weighted-exchangeability class. We approximate the oracle using quantile and kernel histograms, including forest-weighted estimators, and select estimators and tuning parameters by resampling to estimate calibrated target length. On a bounded response domain, we bound excess length linearly in $L^1$ density error at fixed calibration size, without requiring a uniform positive lower bound on the true density. We also bound selection-pool and resampling errors. Under explicit growth conditions, the selected procedure approaches the same oracle while the calibration size remains fixed. Experiments on 36 known-density configurations and four public regression datasets compare coverage, prediction-set length, and runtime with existing methods. In targeted low-coverage and strong-shift comparisons, inverse-density selection yields shorter mean sets in most cases and lower mean runtime than the evaluated weighted baselines, with some differences in achieved coverage.
We give algorithms for approximating the partition function $Z_N$ of the Sherrington--Kirkpatrick (SK) model and finite mixed Ising $p$-spin models with $p\ge 3$. The algorithms apply at a fixed margin below the second-moment threshold $β_{\mathrm f}$ with homogeneous weak external fields $|h|\leqρ_0N^{-1/4}$, for fixed $ρ_0>0$. For any accuracy $\varepsilon\in(0,1)$ and failure parameter $δ\in(0,1)$, they approximate $\log Z_N$ to additive error $\varepsilon$. The number of exact operations is polynomial jointly in $N$, $\varepsilon^{-1}$, and $δ^{-1}$. The disorder variables have i.i.d symmetric distribution with unit variance. The failure probability is at most $δ$ plus an explicit disorder tail bound. This tail bound tends to zero under a finite fourth moment condition. Our approach builds on the cluster expansions of Aizenman--Lebowitz--Ruelle~\cite{ALR87} and Dey--Wu~\cites{DW23,DW26}. For higher-order mixtures, we retain hypergraphs using few vertices and evaluate their total weight deterministically through local spin averages and inclusion--exclusion. For SK, we impose cutoffs on the number of vertices of degree at least three and on the total number of edges. We estimate the retained sum using randomized color coding~\cite{AYZ95}. A lower tail control based on reverse hypercontractivity~\cite{MOS13} converts the mean-square approximation of the normalized partition function into logarithmic accuracy.
We study the scaling of the expected total queue length in an $n\times n$ input-queued switch. It has been conjectured that the optimal scaling is $O(n/(1-ρ))$, up to logarithmic factors, where $ρ$ is the system load. This conjecture has been established in the traditional heavy-traffic regime and in certain sub-regimes of the large-system regime. But it remains unresolved over a wide range of load scalings in the large-system regime. In this paper, we develop a family of scheduling policies parameterized by a positive integer $k\ge 2$. We show that under uniform traffic, our policy achieves an expected total queue length of $O\bigl({n\log\max\{n,1/(1-ρ)\}}\big/{(1-ρ)^{2^k/(2^k-1)}}\bigr)$, as long as $ρ\to 1$ in the large-system regime. As $k$ increases, the exponent $2^k/(2^k-1)$ approaches $1$, bringing our bound arbitrarily close to the conjectured scaling. The expected amortized scheduling cost of the policy is $O(n^3 k)$ per time slot. Our policy is of a batching type, with a novel recursive design that allows an arbitrary number of batching levels. This design principle may also be useful for other scheduling problems in which combinatorial constraints restrict which jobs can be served simultaneously.
Brownian stochastic interpolants connect paired endpoint samples, yet their sampling error can depend on displacement moments even when neither endpoint has a first moment. For arbitrary Borel couplings, we prove early-stopped, target-to-output KL guarantees for standard Euler sampling from the true source. A fourth displacement moment gives an explicit $O(N^{-1})$ bound; a second moment suffices for fixed-law convergence as drift error vanishes. For one fixed coupling with no displacement moment above two, original bridge-path KL stays of order one, comparison-path KL is of order $1/\log N$, and terminal KL is at most $C/N^2$ on the same grid. The path decomposition identifies the information about the source discarded by Markov projection. When the original bridge is Markov, current-state Euler sampling instead has a second-moment $O((M_2+d)/N)$ bound; finite-entropy Schroedinger bridges for the probability Brownian reference are included. We also construct a current-state learner with expected output KL consistency from iid endpoint pairs. Numerical diagnostics illustrate the distinct errors. Conditional diffusion identities specify a restoration interface, while image controls find no iterative advantage over cheap probabilistic predictors. Uniform terminal rates for general second-moment couplings remain open.
It is known that randomly initialized neural networks have Gaussian behaviour in the $n\to \infty$ limit. Unfortunately, this approximation can be poor for finite $n$ and gets worse as one goes deeper into the network. In this work, we develop a more accurate approximation that is \emph{slightly} non-Gaussian, which we call the para-Gaussian distribution and is constructed from the Hermite polynomials. We show how this para-Gaussian approximation can lead to a practical recursive method to approximate the deeper layers of the network. Finally, we apply this theory to linear neural networks (where things can be solved exactly) and to ReLU neural networks (where things can be approximated).
We analyze the expected running time of WalkSAT, a well-known local search procedure for satisfiability solving, on satisfiable instances of the k-XOR SAT problem. We obtain estimates of this expected running time by reducing the problem to a setting amenable to classical techniques from drift analysis. A crucial ingredient of this reduction is the definition of (new, explosive) hypergraph versions of interacting particle systems, notably of coalescing and annihilating random walks as well as the voter model. The use of these tools allows to show that the expected running time of WalkSAT depends on structural parameter (we call odd Cheeger drift) of the dual of the formula hypergraph.
Classical pooled-testing rules choose pool sizes to reduce the number of tests. In practice, however, samples arrive over time, pools take time to form, testing capacity is finite, and positive pools create bursts of follow-up work. These operational features can make the test-efficient pool size far from the delay-efficient one. We formulate pooled testing with stochastic arrivals and finite testing capacity as a queueing system, and derive exact performance representations and asymptotic scaling laws for fixed-batch Dorfman testing and a dynamic two-threshold policy. The analysis reveals three distinct design regimes. When test usage is the main concern, the classical Dorfman rule is recovered and the optimal pool size scales with the inverse square root of prevalence. When delay accumulated while the laboratory is busy testing another batch matters, positive pools create service-time variability that grows more rapidly with pool size than the mean retesting burden, and the optimal scale shifts to the inverse cube root of prevalence. When waiting to assemble a pool is itself costly, the optimal fixed pool size remains bounded even as prevalence becomes very small. Under severe capacity scarcity, congestion remains relevant at the classical Dorfman scale: the square-root law survives, but with a capacity-dependent constant. Dynamic pooling separates two operational decisions that fixed pool sizes conflate. A lower threshold determines when routine testing should begin, while an upper threshold limits pooling when backlog is high. The lower threshold follows the cube-root scale, whereas the upper threshold primarily acts as congestion control. The results show how prevalence, capacity, and delay jointly determine pooling decisions and explain why operationally optimal pools can be much smaller than those suggested by test-efficiency calculations alone.
Despite significant advances in computer vision, semantic segmentation models remain vulnerable to backdoor attacks, posing a security risk. These attacks redirect pixels of a chosen victim class to an attacker-specified target class when a trigger is present, while preserving clean predictions. Existing defenses are largely derived from image classification, while dedicated segmentation defenses remain scarce and typically require retraining or model modification. We introduce RePAIR, a post-hoc, label-free, attack-unaware defense based on the observation that backdoor-induced shifts can be stronger in the classifier head than in internal representations, leaving recoverable victim-class semantics. RePAIR infers the victim-target relation from an unlabeled, potentially contaminated image pool, constructs a safety-approved semantic reference, and localizes suspicious pixels by comparing head predictions with internal semantic evidence. It then applies feature-space and output corrections toward the inferred victim semantics, while conservative localization and output constraints limit changes to benign predictions. We model the attack as adversarial probability transport and repair as its approximate reversal, deriving bounds and sufficient conditions for localization, residual attack success, clean-side changes, defense-induced target predictions, and attack-unaware pair identification. Experiments across segmentation architectures and attack settings show that RePAIR substantially reduces attack success while largely preserving clean performance.
Local-to-Global theorems are a cornerstone of many of recent developments in theoretical computer science from coding theory and PCPs all the way to analyzing mixing time of Markov chains. Building on trickledown theorems on simplicial complexes, several techniques have been established to obtain tighter local-to-global theorems to analyze the spectral gap of the Glauber dynamics, namely matrix trickledown, ${\mathcal C}$-Lorentzian polynomials and more recently Bochner type inequalities. In this paper, we adopt non-local local-to-global/Bochner type inequalities on "cones", corresponding to the local neighborhood of a given vertex, to show that the Glauber dynamics mixes in polynomial time to sample proper colorings of maximum degree $Δ$ graphs as long as $q\geq 1.7408Δ$.