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Jingcheng Liu

Publications and source records attributed to Jingcheng Liu.

At least 19 recordsLinked to original sources
arXiv2026

A Tale of Two Walks: From Kac to KMP and Back

We reveal an unexpected connection between two interacting particle systems: Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. We first establish a mixing-time analogue of Aldous's conjecture for the KMP process on general graphs: the mixing time of the $t$-particle process on any connected $d$-vertex graph is at most $O(\log d)$ times the corresponding single-particle mixing time, uniformly in $t$. As an application, we improve the mixing-time bound for coordinate hit-and-run on the $n$-dimensional standard simplex from $\widetilde{O}(n^3)$ [KV26] to $O(n\log n)$, while removing the dependence on the initial distribution. We then show that the twirling channel induced by the parallel Kac's walk on the symmetric subspace can be exactly encoded by a Markov chain on partitions, which naturally lifts to a parallel KMP chain on the complete graph. We prove that $O(\log d+\log(1/\varepsilon))$ repetitions of the parallel Kac's walk suffice to approximate Haar twirling on the symmetric subspace of $(\mathbb{C}^d)^{\otimes t}$ to error $\varepsilon$, uniformly in all $t$. Our main technical contribution is a new structural property of KMP-type dynamics: conditional product structure, which holds for both the standard KMP processes on general graphs and the parallel KMP processes on complete graphs. Conditioned on suitable auxiliary randomness, the evolution of the labeled KMP process factorizes into independent single-particle updates. Combining this structure with an exact coupling, we remove the dependence on the number of particles from the mixing bounds for both unlabeled KMP models. This bypasses the $\log(π_{\min}^{-1})$ dependency from spectral-gap arguments, which is often prohibitively large for many-particle systems. Using this framework, we prove mixing bounds that are sharp up to logarithmic factors for both unlabeled KMP models.

quant-ph↗
arXiv2026

Fine-Grained Complexity Across the Computational Phase Transition

How does a computational problem transition from easy to hard? In the hard-core model, approximate counting and sampling are polynomial-time tractable up to tree uniqueness, but worst-case intractable at any constant relative distance above it unless $\mathsf{NP}=\mathsf{RP}$. We quantify the transition from polynomial to exponential mixing for classical Glauber dynamics. On $n$-vertex graphs of maximum degree $d\ge3$, at fugacity $λ=(1+δ)λ_{\mathrm c}(d)$ with sufficiently small $δ\ge0$ that may vanish with $n$, its worst-case mixing time has the form $\exp\{Θ(nδ^2/d)\}n^{1+Θ(1)}$. The upper bound is degree-uniform, and the matching lower bound holds for every fixed $d$. Annealing gives approximate counting with the same running-time upper bound at fixed accuracy. For general sampling and counting algorithms, we establish a robust fine-grained reduction from SAT without assuming a lower bound on SAT's running time. For any fixed $d\ge3$ and $0<γ<1$, a sampling or approximate-counting algorithm running in $\exp\{O((nδ^2/d)^{1-γ})\}\operatorname{poly}(n)$ time along any one suitable inverse-polynomial distance sequence $δ=δ(n)$, yields a randomized SAT algorithm running in $\exp\{\widetilde O(s^{1-γ})\}$ time on inputs of binary length $s$. This power-saving transfer identifies $nδ^2/d$ as the analogue of SAT input length for hard-core sampling and counting at relative distance $δ$ above criticality: beating the Glauber exponent by a fixed power through any sampling or counting method would yield a corresponding power improvement for SAT. At criticality, we establish $O(\sqrt{n/d})$ spectral independence, optimal in $n$ at fixed degree, and degree-uniform Glauber mixing in $\widetilde O(n(n/d)^{1+1/(d-2)})$ time, improving the previous $\widetilde O(n^{4+4/(d-2)})$ bound.

cs.CC↗
arXiv2026

Human-Supervised Artificial Intelligence Researchers in Electric Power Systems: Knowledge, Physics, Simulation, and Governance for Large Language Models

Conclusions in power-system research require a clear evidential basis. Researchers need to understand the data sources, modelling assumptions, and solution procedures to assess the conditions under which a conclusion holds. Large language models (LLMs) can help researchers extract information from documents and translate research requests into code and tool calls, but their outputs still require physical validation. Focusing on human-supervised artificial intelligence (AI) researchers, this review examines power-system LLM applications and related scientific workflows, together with physics-grounded power AI studies, to discuss the conditions for using LLMs in scientific research. In the proposed knowledge-physics-simulation-governance framework, LLMs translate information and coordinate tasks, simulators and validation tools test models, and experts retain authority over critical assumptions, safety boundaries, and final conclusions. Research processes should preserve materials for review so that claims can be traced to their sources, computations can be reproduced, and failed attempts and approvals can be examined. On this basis, a staged development roadmap and requirements for research reporting are proposed. Evaluation should focus on research artifacts and their validation to assess whether LLMs can improve the efficiency of power-system research within clearly defined permissions.

cs.AI↗
arXiv2026

Simulated annealing and Weak Poincaré inequalities with inverse-polynomial accuracy for the Sherrington-Kirkpatrick model

We study sampling from the Gibbs distribution of the Sherrington-Kirkpatrick (SK) model with Glauber dynamics. For every fixed inverse temperature $0\leqβ<1$ and every fixed $M,D>0$, we give a polynomial-time simulated annealing algorithm whose output distribution is within $n^{-M}$ total-variation distance of the Gibbs distribution, with probability at least $1-n^{-D}$ over the interaction matrix. The algorithm starts from uniform product spins and uses Glauber dynamics along an increasing inverse-temperature schedule. The same approach also yields partition-function estimates with relative error $n^{-M}$. The key ingredient is a quantitative weak Poincaré inequality. Building on the stochastic-localization approach to weak Poincaré inequalities [Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026], we use Gaussian isoperimetry to directly compare exact stochastic localization paths. Our comparison tolerates failures of local Lipschitz continuity of the posterior mean by controlling their accumulated cost. The local Lipschitzness comes from approximating the posterior means by stationary points of the Thouless-Anderson-Palmer (TAP) free energy in locally strongly convex regions located by approximate message passing, a paradigm introduced by algorithmic stochastic localization [El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]. The approximation errors are roughly characterized by the validity of the TAP gradient equations, but in a different coordinate. To show that the errors rarely accumulate too much, we need a strong concentration control. A direct argument would require solving a variant of an open problem by Talagrand [2010, Research Problem 1.7.9]. We bypass this open problem with an intermediate conditioning step and transfer the control back by establishing the stability of the TAP gradient under deletion of coordinates.

cs.DS↗
arXiv2026

The Price of Privacy For Approximating Max-CSP

We study approximation algorithms for Maximum Constraint Satisfaction Problems (Max-CSPs) under differential privacy (DP) where the constraints are considered sensitive data. Information-theoretically, we aim to classify the best approximation ratios possible for a given privacy budget $\varepsilon$. In the high-privacy regime ($\varepsilon \ll 1$), we show that any $\varepsilon$-DP algorithm cannot beat a random assignment by more than $O(\varepsilon)$ in the approximation ratio. We devise a polynomial-time algorithm which matches this barrier under the assumptions that the instances are bounded-degree and triangle-free. Finally, we show that one or both of these assumptions can be removed for specific CSPs--such as Max-Cut or Max $k$-XOR--albeit at the cost of computational efficiency.

cs.DS↗
arXiv2026

Private Approximation of Graph Spectra and Cuts via Spectral Amplifiers

We study the problem of releasing a synthetic graph that approximates the sizes of all cuts of an input graph under edge-level differential privacy. If one insists on purely additive error, the optimal worst-case error is $\widetildeΘ(n^{3/2})$. If one allows a small multiplicative slack, an information-theoretic exponential-time mechanism achieves nearly linear additive error, but the best known polynomial-time algorithms have substantially larger error. We give a polynomial-time $(\varepsilon,δ)$-differentially private algorithm which, for every $n$-vertex unweighted graph $G$, outputs a non-negative weighted synthetic graph $\widetilde G$ such that, with high probability, every cut $S\subseteq V(G)$ satisfies \[ |w_G(S)-w_{\widetilde G}(S)| \le γw_G(S)+\widetilde O_{\varepsilon,δ,γ}(n^{13/12+o(1)}). \] This improves the previous polynomial-time worst-case bound $\widetilde O(n^{5/4+o(1)})$ of Aamand et al. (ICML 2025) for mixed multiplicative/additive private cut approximation. The main technical ingredient is a new set of private spectral primitives for bounded-degree graphs, one of them gives spectral error $\widetilde O_δ((nd)^{1/4}/\sqrt\varepsilon)$ in estimating the graph Laplacian for graphs of maximum degree $d$, being the first to beat the standard $\min\{2d,\widetilde O_δ(\sqrt{n}/\varepsilon)\}$ baseline in the high-degree regime. We further develop a primitive with a sharper error dependence on $n$ and $d$ for the downstream cut approximation. Combined with a new edge-sensitive terminal cut oracle with additive error $\widetilde O(n+(n^2M)^{1/3})$ on graphs with $M$ edges, this yields the final worst-case $\widetilde O(n^{13/12+o(1)})$ private cut-release error.

cs.DS↗
arXiv2026

Recovery thresholds for hidden weighted sparse graphs

Recovering structural information from noisy high-dimensional data is a fundamental task in statistical inference. We investigate the recovery thresholds for a graph hidden in a randomly weighted complete graph. Specifically, an unknown graph $H^* \in H_n$ is chosen uniformly at random, and hidden in a complete graph of $n$ vertices as follows: the weight of an edge $e \in H$ is distributed independently according to $P_n$; otherwise the weight is distributed independently according to $Q_n$. The goal is to recover almost all of $H$ from these edge weights. Assuming a local Lipschitzness of the Rényi divergence between distributions $P_n$ and $Q_n$, and a mild density condition for the graphs $H_n$, we give a unified characterization of the information-theoretic limit for recovering almost all of $H$ (also known as almost exact recovery). Our characterization connects the KL divergence between $P_n$ and $Q_n$ to the logarithm of the first moment threshold of $H$ in the Erdős-Rényi random graph model $G(n,p)$. Our lower bound also extends to the task of partial recovery, in which only a constant $λ$-fraction of $H$ needs to be recovered. Last but not least, for certain Bernoulli and Exponential regimes, and for Gaussian distributions, we are able to show an All-or-Nothing (AoN) threshold phenomenon at the exponential scale.

math.ST↗
arXiv2026

Zero-free regions and concentration inequalities for hypergraph colorings in the local lemma regime

We show that for $q$-colorings in $k$-uniform hypergraphs with maximum degree $Δ$, if $k\ge 50$ and $q\ge 700Δ^{\frac{5}{k-10}}$, there is a "Lee-Yang" zero-free strip around the interval $[0,1]$ of the partition function, which includes the special case of uniform enumeration of hypergraph colorings. As an immediate consequence, we obtain Berry-Esseen type inequalities for hypergraph $q$-colorings under such conditions, demonstrating the asymptotic normality for the size of any color class in a uniformly random coloring. Our framework also extends to the study of "Fisher zeros", leading to deterministic algorithms for approximating the partition function in the zero-free region. Our approach is based on extending the recent work of [Liu, Wang, Yin, Yu, STOC 2025] to general constraint satisfaction problems (CSP). We focus on partition functions defined for CSPs by introducing external fields to the variables. A key component in our approach is a projection-lifting scheme, which enables us to essentially lift information percolation type analysis for Markov chains from the real line to the complex plane. Last but not least, we also show a Chebyshev-type inequality under the sampling LLL condition for atomic CSPs.

cs.DS↗
arXiv2025

Optimal quantum sampling on distributed databases

Quantum sampling, a fundamental subroutine in numerous quantum algorithms, involves encoding a given probability distribution in the amplitudes of a pure state. Given the hefty cost of large-scale quantum storage, we initiate the study of quantum sampling in a distributed setting. Specifically, we assume that the data is distributed among multiple machines, and each machine solely maintains a basic oracle that counts the multiplicity of individual elements. Given a quantum sampling task, which is to sample from the joint database, a coordinator can make oracle queries to all machines. We focus on the oblivious communication model, where communications between the coordinator and the machines are predetermined. We present both sequential and parallel algorithms: the sequential algorithm queries the machines sequentially, while the parallel algorithm allows the coordinator to query all machines simultaneously. Furthermore, we prove that both algorithms are optimal in their respective settings.

quant-ph↗
arXiv2025

A Generalized Binary Tree Mechanism for Differentially Private Approximation of All-Pair Distances

We study the problem of approximating all-pair distances in a weighted undirected graph with differential privacy, introduced by Sealfon [Sea16]. Given a publicly known undirected graph, we treat the weights of edges as sensitive information, and two graphs are neighbors if their edge weights differ in one edge by at most one. We obtain efficient algorithms with significantly improved bounds on a broad class of graphs which we refer to as \textit{recursively separable}. In particular, for any $n$-vertex $K_h$-minor-free graph, our algorithm achieve an additive error of $\widetilde{O}(h(nW)^{1/3} ) $, where $ W $ represents the maximum edge weight; For grid graphs, the same algorithmic scheme achieve additive error of $\widetilde{O}(n^{1/4}\sqrt{W})$. Our approach can be seen as a generalization of the celebrated binary tree mechanism for range queries, as releasing range queries is equivalent to computing all-pair distances on a path graph. In essence, our approach is based on generalizing the binary tree mechanism to graphs that are \textit{recursively separable}.

cs.DS↗
arXiv2025

Phase Transitions via Complex Extensions of Markov Chains

We study algebraic properties of partition functions, particularly the location of zeros, through the lens of rapidly mixing Markov chains. The classical Lee-Yang program initiated the study of phase transitions via locating complex zeros of partition functions. Markov chains, besides serving as algorithms, have also been used to model physical processes tending to equilibrium. In many scenarios, rapid mixing of Markov chains coincides with the absence of phase transitions (complex zeros). Prior works have shown that the absence of phase transitions implies rapid mixing of Markov chains. We reveal a converse connection by lifting probabilistic tools for the analysis of Markov chains to study complex zeros of partition functions. Our motivating example is the independence polynomial on $k$-uniform hypergraphs, where the best-known zero-free regime has been significantly lagging behind the regime where we have rapidly mixing Markov chains for the underlying hypergraph independent sets. Specifically, the Glauber dynamics is known to mix rapidly on independent sets in a $k$-uniform hypergraph of maximum degree $Δ$ provided that $Δ\lesssim 2^{k/2}$. On the other hand, the best-known zero-freeness around the point $1$ of the independence polynomial on $k$-uniform hypergraphs requires $Δ\le 5$, the same bound as on a graph. By introducing a complex extension of Markov chains, we lift an existing percolation argument to the complex plane, and show that if $Δ\lesssim 2^{k/2}$, the Markov chain converges in a complex neighborhood, and the independence polynomial itself does not vanish in the same neighborhood. In the same regime, our result also implies central limit theorems for the size of a uniformly random independent set, and deterministic approximation algorithms for the number of hypergraph independent sets of size $k \le αn$ for some constant $α$.

cs.DS↗
arXiv2024

Perfect sampling from rapidly mixing Markov chains

We show that efficient approximate sampling algorithms, combined with a slow exponential time oracle for computing its output distribution, can be combined into constructing efficient perfect samplers, which sample exactly from a target distribution with zero error upon termination. This extends a classical reduction of Jerrum, Valiant and Vazirani, which says that for self-reducible problems, deterministic approximate counting can be used to construct perfect samplers. We provide two surprisingly simple constructions, and our perfect samplers run in polynomial time both in expectation and with high probability. An overwhelming amount of efficient approximate sampling algorithms are based on Markov chains. Informally, we show that any Markov chains with absolute spectral gap $γ$ can be converted into a perfect sampler with expected time $O\left(\frac{1}γ\ln\frac{|Ω|}{π_{*}}\right)$, where $π_{*}$ is the minimum probability in the stationary distribution. This is also the best possible bound for mixing time to achieve approximate sampling from a spectral gap, and we are able to do perfect sampling in the same time bound in expectation. We also highlight a number of applications where we either get the first perfect sampler up to the uniqueness regime (roughly speaking, everywhere except where NP-hardness results are known), or the fastest perfect sampler known to date. Remarkably, we are able to get the first perfect sampler for perfect matchings of bipartite graphs based on the celebrated Jerrum-Sinclair-Vigoda algorithm.

cs.CC↗
arXiv2024

Optimality of Matrix Mechanism on $\ell_p^p$-metric

In this paper, we introduce the $\ell_p^p$-error metric (for $p \geq 2$) when answering linear queries under the constraint of differential privacy. We characterize such an error under $(ε,δ)$-differential privacy. Before this paper, tight characterization in the hardness of privately answering linear queries was known under $\ell_2^2$-error metric (Edmonds et al., STOC 2020) and $\ell_p^2$-error metric for unbiased mechanisms (Nikolov and Tang, ITCS 2024). As a direct consequence of our results, we give tight bounds on answering prefix sum and parity queries under differential privacy for all constant $p$ in terms of the $\ell_p^p$ error, generalizing the bounds in Henzinger et al. (SODA 2023) for $p=2$.

cs.CR↗
arXiv2024

Almost linear time differentially private release of synthetic graphs

In this paper, we give an almost linear time and space algorithms to sample from an exponential mechanism with an $\ell_1$-score function defined over an exponentially large non-convex set. As a direct result, on input an $n$ vertex $m$ edges graph $G$, we present the \textit{first} $\widetilde{O}(m)$ time and $O(m)$ space algorithms for differentially privately outputting an $n$ vertex $O(m)$ edges synthetic graph that approximates all the cuts and the spectrum of $G$. These are the \emph{first} private algorithms for releasing synthetic graphs that nearly match this task's time and space complexity in the non-private setting while achieving the same (or better) utility as the previous works in the more practical sparse regime. Additionally, our algorithms can be extended to private graph analysis under continual observation.

cs.CR↗
arXiv2023

Optimal Bounds on Private Graph Approximation

We propose an efficient $ε$-differentially private algorithm, that given a simple {\em weighted} $n$-vertex, $m$-edge graph $G$ with a \emph{maximum unweighted} degree $Δ(G) \leq n-1$, outputs a synthetic graph which approximates the spectrum with $\widetilde{O}(\min\{Δ(G), \sqrt{n}\})$ bound on the purely additive error. To the best of our knowledge, this is the first $ε$-differentially private algorithm with a non-trivial additive error for approximating the spectrum of the graph. One of the subroutines of our algorithm also precisely simulates the exponential mechanism over a non-convex set, which could be of independent interest given the recent interest in sampling from a {\em log-concave distribution} defined over a convex set. Spectral approximation also allows us to approximate all possible $(S,T)$-cuts, but it incurs an error that depends on the maximum degree, $Δ(G)$. We further show that using our sampler, we can also output a synthetic graph that approximates the sizes of all $(S,T)$-cuts on $n$ vertices weighted graph $G$ with $m$ edges while preserving $(ε,δ)$-differential privacy and an additive error of $\widetilde{O}(\sqrt{mn}/ε)$. We also give a matching lower bound (with respect to all the parameters) on the private cut approximation for weighted graphs. This removes the gap of $\sqrt{W_{\mathsf{avg}}}$ in the upper and lower bound in Eli{á}{š}, Kapralov, Kulkarni, and Lee (SODA 2020), where $W_{\mathsf{avg}}$ is the average edge weight.

cs.DS↗
arXiv2023

Uniqueness and Rapid Mixing in the Bipartite Hardcore Model

We characterize the uniqueness condition in the hardcore model for bipartite graphs with degree bounds only on one side, and provide a nearly linear time sampling algorithm that works up to the uniqueness threshold. We show that the uniqueness threshold for bipartite graph has almost the same form of the tree uniqueness threshold for general graphs, except with degree bounds only on one side of the bipartition. The hardcore model from statistical physics can be seen as a weighted enumeration of independent sets. Its bipartite version (#BIS) is a central open problem in approximate counting. Compared to the same problem in a general graph, surprising tractable regime have been identified that are believed to be hard in general. This is made possible by two lines of algorithmic approach: the high-temperature algorithms starting from Liu and Lu (STOC 2015), and the low-temperature algorithms starting from Helmuth, Perkins, and Regts (STOC 2019). In this work, we study the limit of these algorithms in the high-temperature case. Our characterization of the uniqueness condition is obtained by proving decay of correlations for arguably the best possible regime, which involves locating fixpoints of multivariate iterative rational maps and showing their contraction. We also give a nearly linear time sampling algorithm based on simulating field dynamics only on one side of the bipartite graph that works up to the uniqueness threshold. Our algorithm is very different from the original high-temperature algorithm of Liu and Lu, and it makes use of a connection between correlation decay and spectral independence of Markov chains. Last but not the least, we are able to show that the standard Glauber dynamics on both side of the bipartite graph mixes in polynomial time up to the uniqueness.

cs.DS↗
arXiv2022

Correlation decay and partition function zeros: Algorithms and phase transitions

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $Δ$-regular tree) implies strong spatial mixing on all graphs of maximum degree $Δ$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of $q$-colorings of a graph of maximum degree $Δ$ provided only that $q\ge 2Δ$. This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs.

cs.CC↗
arXiv2019

Zeros of ferromagnetic 2-spin systems

We study zeros of the partition functions of ferromagnetic 2-state spin systems in terms of the external field, and obtain new zero-free regions of these systems via a refinement of Asano's and Ruelle's contraction method. The strength of our results is that they do not depend on the maximum degree of the underlying graph. Via Barvinok's method, we also obtain new efficient and deterministic approximate counting algorithms. In certain regimes, our algorithm outperforms all other methods such as Markov chain Monte Carlo and correlation decay.

cs.DS↗