We develop a framework for deterministic approximate counting of multi-spin systems beyond bounded-degree graphs. The algorithm recursively constructs rational polytopes containing the true marginal vectors and uses linear-fractional programming to obtain certified bounds on marginal ratios. For positive interactions on graphs of polynomial connective constant $D$, we establish strong spatial mixing and a fully polynomial-time approximation scheme (\textbf{FPTAS}) whenever $Dc<1$, where $c$ bounds the Birkhoff contraction coefficients of the interactions. We further extend the framework to proper colorings of sparse Erd\H{o}s-R\'{e}nyi random graphs using recursion on permissive blocks. For every fixed $\eta\in(0,1)$, sufficiently large fixed $d$, and fixed integer $q\ge(2+\eta)d$, we obtain an \textbf{FPTAS} for counting proper $q$-colorings of $G\sim\mathcal G(n,d/n)$ with high probability over $G$. This improves the leading constant $3$ in the earlier counting guarantee of Yin and Zhang (APPROX/RANDOM, 2016) to $2$, and asymptotically matches the spatial mixing regime established by Yin (ICALP, 2014).
We give deterministic FPTASes for two dense counting problems on which the known deterministic algorithms, based on zero-free interpolation, run in quasipolynomial time. For fixed $0<\gamma<1/2$ and $0<\theta\leq1$, the first approximates $\mathrm{haf}(A)$ for a symmetric matrix $A$ when its support graph $G$ has minim...
We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps...
We study deterministic relative approximation of the total variation distance between two Gibbs distributions induced by spin systems on the same bounded-degree graph. For the hard-core model, we give a deterministic polynomial-time $\varepsilon$-relative-error approximation when both external-field vectors lie in $[b,...
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
Assuming the Exponential Time Hypothesis (ETH), fixed-parameter tractability and polynomial-time solvability coincide for homomorphism problems specified by classes of pattern hypergraphs of bounded incidence degeneracy or bounded primal independence number. In both cases, tractability is characterised by bounded fract...
We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to...
En-Kai Zhang· 0 citations
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