Absence of zeros implies strong spatial mixing

Regts, Guus

doi:10.1007/s00440-023-01190-z

Cited by 2 publications

(2 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To approximate (), we use the method of polynomial interpolation, see [4, 25] as general references, as well as recent [9, 12, 28] for connections with other computational approaches, correlation decay and Markov chain Monte Carlo. For that, we construct a polynomial

{P}_{A,b,\gamma}\left(\mathbf{z}\right)=\sum \limits_{\xi_1,\dots, {\xi}_n\in \left\{0,1\right\}}{z}_1^{\xi_1}\cdots {z}_n^{\xi_n}\exp \left\{-\sum \limits_{i=1}^n{\gamma}_i{\left(-{\beta}_i+\sum \limits_{j=1}^n{\alpha}_{ij}{\xi}_j\right)}^2\right\},

in an

n

‐vector

\mathbf{z}=\left({z}_1,\dots, {z}_n\right)

of complex variables.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Smoothed counting of 0–1 points in polyhedra

Barvinok

2022

Random Struct Algorithms

View full text Add to dashboard Cite

Given a system of linear equations ℓifalse(xfalse)=βi$$ {\ell}_i(x)={\beta}_i $$ in an n$$ n $$‐vector x$$ x $$ of 0–1 variables, we compute the expectation of exp{}prefix−∑iγi()ℓifalse(xfalse)prefix−βi2$$ \exp \left\{-{\sum}_i{\gamma}_i{\left({\ell}_i(x)-{\beta}_i\right)}^2\right\} $$, where x$$ x $$ is a vector of independent Bernoulli random variables and γi>0$$ {\gamma}_i>0 $$ are constants. The algorithm runs in quasi‐polynomial nOfalse(lnnfalse)$$ {n}^{O\left(\ln n\right)} $$ time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. We discuss applications to perfect matchings in hypergraphs and randomized rounding in discrete optimization.

show abstract

{P}_{A,b,\gamma}\left(\mathbf{z}\right)=\sum \limits_{\xi_1,\dots, {\xi}_n\in \left\{0,1\right\}}{z}_1^{\xi_1}\cdots {z}_n^{\xi_n}\exp \left\{-\sum \limits_{i=1}^n{\gamma}_i{\left(-{\beta}_i+\sum \limits_{j=1}^n{\alpha}_{ij}{\xi}_j\right)}^2\right\},

in an

n

‐vector

\mathbf{z}=\left({z}_1,\dots, {z}_n\right)

of complex variables.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Smoothed counting of 0–1 points in polyhedra

Barvinok

2022

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

“…(1.7) Other applications. Zero-free regions of partition functions of the type covered by Theorem 1.1 turn out to be relevant to the decay of correlations [Ga23], [Re23], to the mixing time of Markov Chains [Ch+22], to the validity of the Central Limit Theorem for combinatorial structures [MS19], as well as to other related algorithmic applications [J+22].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

More on zeros and approximation of the Ising partition function

Barvinok

2021

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We consider the problem of computing the partition function $\sum _x e^{f(x)}$ , where $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$ . In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln \epsilon )}$ time if the Lipschitz constant of the non-linear part of f with respect to the $\ell ^1$ metric on the Boolean cube does not exceed $1-\delta $ , for any $\delta>0$ , fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that $\sum _x e^{\tilde {f}(x)} \ne 0$ for complex-valued polynomials $\tilde {f}$ in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices.

show abstract

Absence of zeros implies strong spatial mixing

Cited by 2 publications

References 41 publications

Smoothed counting of 0–1 points in polyhedra

Smoothed counting of 0–1 points in polyhedra

More on zeros and approximation of the Ising partition function

Contact Info

Product

Resources

About