arXiv subjects

Alexander Barvinok

Publications and source records attributed to Alexander Barvinok.

At least 19 recordsLinked to original sources

arXiv2026

On the dependence of the zero-free region of a partition function on the external field

Let $\{0, 1\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\Bbb P}(1)=p$ and ${\Bbb P}(0)=q$ with $0 < p \leq q$ and $p+q=1$. For $i=1, \ldots, m$, let $\phi_i: \{0, 1\}^n \longrightarrow {\Bbb C}$ be $L_i$-Lipschitz functions in the Hamming metric, such that each $\phi_i$ depends on at most $r$ coordinates of $x \in \{0, 1\}^n$, where $rp \geq 12$. For $j=1, \ldots, n$, let $I_j $ be the set of indices $i$ such that $\phi_i$ depends on the $j$-th coordinate. We prove that $E \exp\left\{ \sum_{i=1}^m \phi_i \right\} \ne 0$ provided $\sum_{i \in I_j} L_i \leq {1 \over 10 \sqrt{rp}}$ for all $j$. This translates into a regime for $\pm 1$ spin systems, where a linear increase in the energy of multi-spin interactions requires only a logarithmic increase of the external field to keep the partition function zero-free and the system away from the phase transition. As a corollary, we obtain efficient deterministic algorithms to approximate the partition function in the zero-free region.

math-ph↗

arXiv2026

Computing Gaussian and exponential integrals in ${\Bbb R}^n$

We consider expectations of the type $E \exp \left\{\sum_{i=1}^m \phi_i \right\}$, where $\phi_i: {\Bbb R}^n \longrightarrow {\Bbb C}$ are functions, each depending on a few coordinates of a point in ${\Bbb R}^n$, and the expectation is taken with respect to the standard Gaussian or symmetric exponential probability measures. We prove sufficient conditions, in terms of the Lipschitz constants of $\phi_i$ and the combinatorics of their dependencies, for the integral to be non-zero, and, consequently, to be amenable to a computationally efficient approximation. We discuss applications to computing volumes of bodies and statistics on integer points in polyhedra in ${\Bbb R}^n$.

cs.DS↗

arXiv2025

Computing the probability of intersection

Let $\Omega_1, \ldots, \Omega_m$ be probability spaces, let $\Omega=\Omega_1 \times \cdots \times \Omega_m$ be their product and let $A_1, \ldots, A_n \subset \Omega$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x \in \Omega$, $x=\left(\xi_1, \ldots, \xi_m\right)$, and that for each event $A_i$ there are $\Delta_i$ of other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $\Delta=\max\{5,\ \Delta_i: i=1, \ldots, n\}$ and let $\mu_i=\min\{r_i,\ \Delta_i+1\}$ for $i=1, \ldots, n$. We prove that if $P(A_i) < (3\Delta)^{-3\mu_i}$ for all $i$, then for any $0 < \epsilon < 1$, the probability $P\left( \bigcap_{i=1}^n \overline{A}_i\right)$ of the intersection of the complements of all $A_i$ can be computed within relative error $\epsilon$ in polynomial time from the probabilities $P\left(A_{i_1} \cap \ldots \cap A_{i_k}\right)$ of $k$-wise intersections of the events $A_i$ for $k = e^{O(\Delta)} \ln (n/\epsilon)$.

math.PR↗

arXiv2024

On the zeros of partition functions with multi-spin interactions

Let $X_1, \ldots, X_n$ be probability spaces, let $X$ be their direct product, let $\phi_1, \ldots, \phi_m: X \longrightarrow {\Bbb C}$ be random variables, each depending only on a few coordinates of a point $x=(x_1, \ldots, x_n)$, and let $f=\phi_1 + \ldots + \phi_m$. The expectation $E\thinspace e^{\lambda f}$, where $\lambda \in {\Bbb C}$, appears in statistical physics as the partition function of a system with multi-spin interactions, and also in combinatorics and computer science, where it is known as the partition function of edge-coloring models, tensor network contractions or a Holant polynomial. Assuming that each $\phi_i$ is 1-Lipschitz in the Hamming metric of $X$, that each $\phi_i(x)$ depends on at most $r \geq 2$ coordinates $x_1, \ldots, x_n$ of $x \in X$, and that for each $j$ there are at most $c \geq 1$ functions $\phi_i$ that depend on the coordinate $x_j$, we prove that $E\thinspace e^{\lambda f} \ne 0$ provided $| \lambda | \leq \ (3 c \sqrt{r-1})^{-1}$ and that the bound is sharp up to a constant factor. Taking a scaling limit, we prove a similar result for functions $\phi_1, \ldots, \phi_m: {\Bbb R}^n \longrightarrow {\Bbb C}$ that are 1-Lipschitz in the $\ell^1$ metric of ${\Bbb R}^n$ and where the expectation is taken with respect to the standard Gaussian measure in ${\Bbb R}^n$. As a corollary, the value of the expectation can be efficiently approximated, provided $\lambda$ lies in a slightly smaller disc.

math.PR↗

arXiv2023

Computing the theta function

Let $f: {\Bbb R}^n \longrightarrow {\Bbb R}$ be a positive definite quadratic form and let $y \in {\Bbb R}^n$ be a point. We present a fully polynomial randomized approximation scheme (FPRAS) for computing $\sum_{x \in {\Bbb Z}^n} e^{-f(x)}$, provided the eigenvalues of $f$ lie in the interval roughly between $s$ and $e^{s}$ and for computing $\sum_{x \in {\Bbb Z}^n} e^{-f(x-y)}$, provided the eigenvalues of $f$ lie in the interval roughly between $e^{-s}$ and $s^{-1}$ for some $s \geq 3$. To compute the first sum, we represent it as the integral of an explicit log-concave function on ${\Bbb R}^n$, and to compute the second sum, we use the reciprocity relation for theta functions. We then apply our results to test the existence of many short integer vectors in a given subspace $L \subset {\Bbb R}^n$, to estimate the distance from a given point to a lattice, and to sample a random lattice point from the discrete Gaussian distribution.

math.NA↗

arXiv2022

A quick estimate for the volume of a polyhedron

Let $P$ be a bounded polyhedron defined as the intersection of the non-negative orthant ${\Bbb R}^n_+$ and an affine subspace of codimension $m$ in ${\Bbb R}^n$. We show that a simple and computationally efficient formula approximates the volume of $P$ within a factor of $γ^m$, where $γ>0$ is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.

math.MG↗

arXiv2021

When a system of real quadratic equations has a solution

We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite program, one can reduce it to another system of the type $q_i(x)=α_i$, $i=1, \ldots, m$, where $q_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms and $α_i=\mathrm{tr\ } q_i$. We prove that the latter system has solution $x \in {\mathbb R}^n$ if for some (equivalently, for any) orthonormal basis $A_1,\ldots, A_m$ in the space spanned by the matrices of the forms $q_i$, the operator norm of $A_1^2 + \ldots + A_m^2$ does not exceed $η/m$ for some absolute constant $η> 0$. The condition can be checked in polynomial time and is satisfied, for example, for random $q_i$ provided $m \leq γ\sqrt{n}$ for an absolute constant $γ>0$. We prove a similar sufficient condition for a system of homogeneous quadratic equations to have a non-trivial solution. While the condition we obtain is of an algebraic nature, the proof relies on analytic tools including Fourier analysis and measure concentration.

math.OC↗

arXiv2021

Smoothed counting of 0-1 points in polyhedra

Given a system of linear equations $\ell_i(x)=β_i$ in an $n$-vector $x$ of 0-1 variables, we compute the expectation of $\exp\left\{- \sum_i γ_i \left(\ell_i(x) - β_i\right)^2\right\}$, where $x$ is a vector of independent Bernoulli random variables and $γ_i >0$ are constants. The algorithm runs in quasi-polynomial $n^{O(\ln n)}$ 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.

math.CO↗

arXiv2021

More on zeros and approximation of the Ising partition function

We consider the problem of computing the partition function $\sum_x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ 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 $0 < ε< 1$ in quasi-polynomial $n^{O(\ln n - \ln ε)}$ 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 $1-δ$, for any $δ>0$, 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.

math.PR↗

arXiv2020

Testing systems of real quadratic equations for approximate solutions

Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from systems that are far from having a solution. For that, we pick a delta-shaped penalty function $F: {\Bbb R} \longrightarrow [0, 1]$ with $F(0)=1$ and $F(y) < 1$ for $y \ne 0$ and compute the expectation of $F(q_1(x)) \cdots F(q_m(x))$ for a random $x$ sampled from the standard Gaussian measure in ${\Bbb R}^n$. We choose $F(y)=y^{-2}\sin^2 y$ and show that the expectation can be approximated within relative error $0< ε< 1$ in quasi-polynomial time $(m+n)^{O(\ln (m+n)-\ln ε)}$, provided each form $q_i$ depends on not more than $r$ real variables, has common variables with at most $r-1$ other forms and satisfies $|q_i(x)| \leq γ\|x\|^2/r$, where $γ>0$ is an absolute constant. This allows us to distinguish between "easily solvable" and "badly unsolvable" systems in some non-trivial situations.

math.OC↗

arXiv2020

A remark on approximating permanents of positive definite matrices

Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of $A$.

cs.DS↗

arXiv2020

Integrating products of quadratic forms

We prove that if $q_1, \ldots, q_m: {\Bbb R}^n \longrightarrow {\Bbb R}$ are quadratic forms in variables $x_1, \ldots, x_n$ such that each $q_k$ depends on at most $r$ variables and each $q_k$ has common variables with at most $r$ other forms, then the average value of the product $\left(1+ q_1\right) \cdots \left(1+q_m\right)$ with respect to the standard Gaussian measure in ${\Bbb R}^n$ can be approximated within relative error $ε>0$ in quasi-polynomial $n^{O(1)} m^{O(\ln m -\ln ε)}$ time, provided $|q_k(x)| \leq γ\|x\|^2 /r$ for some absolute constant $γ> 0$ and $k=1, \ldots, m$. When $q_k$ are interpreted as pairwise squared distances for configurations of points in Euclidean space, the average can be interpreted as the partition function of systems of particles with mollified logarithmic potentials. We sketch a possible application to testing the feasibility of systems of real quadratic equations.

math.PR↗

arXiv2019

Stability and complexity of mixed discriminants

We show that the mixed discriminant of $n$ positive semidefinite $n \times n$ real symmetric matrices can be approximated within a relative error $ε>0$ in quasi-polynomial $n^{O(\ln n -\ln ε)}$ time, provided the distance of each matrix to the identity matrix in the operator norm does not exceed some absolute constant $γ_0 >0$. We deduce a similar result for the mixed discriminant of doubly stochastic $n$-tuples of matrices from the Marcus - Spielman - Srivastava bound on the roots of the mixed characteristic polynomial. Finally, we construct a quasi-polynomial algorithm for approximating the sum of $m$-th powers of principal minors of a matrix, provided the operator norm of the matrix is strictly less than 1. As is shown by Gurvits, for $m=2$ the problem is $\#P$-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2.

cs.DS↗

arXiv2019

Weighted counting of solutions to sparse systems of equations

Given complex numbers $w_1, \ldots, w_n$, we define the weight $w(X)$ of a set $X$ of 0-1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors $(x_1, \ldots, x_n)$ in $X$. We present an algorithm, which for a set $X$ defined by a system of homogeneous linear equations with at most $r$ variables per equation and at most $c$ equations per variable, computes $w(X)$ within relative error $ε>0$ in $(rc)^{O(\ln n-\ln ε)}$ time provided $|w_j| \leq β(r \sqrt{c})^{-1}$ for an absolute constant $β>0$ and all $j=1, \ldots, n$. A similar algorithm is constructed for computing the weight of a linear code over ${\Bbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

math.CO↗

arXiv2018

Computing permanents of complex diagonally dominant matrices and tensors

We prove that for any $λ> 1$, fixed in advance, the permanent of an $n \times n$ complex matrix, where the absolute value of each diagonal entry is at least $λ$ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error $0 < ε< 1$ in quasi-polynomial $n^{O(\ln n - \ln ε)}$ time. We extend this result to multidimensional permanents of tensors and discuss its application to weighted counting of perfect matchings in hypergraphs.

math.CO↗

arXiv2018

Searching for dense subsets in a graph via the partition function

For a set $S$ of vertices of a graph $G$, we define its density $0 \leq σ(S) \leq 1$ as the ratio of the number of edges of $G$ spanned by the vertices of $S$ to ${|S| \choose 2}$. We show that, given a graph $G$ with $n$ vertices and an integer $m$, the partition function $\sum_S \exp\{ γm σ(S) \}$, where the sum is taken over all $m$-subsets $S$ of vertices and $0 < γ<1$ is fixed in advance, can be approximated within relative error $0 < ε< 1$ in quasi-polynomial $n^{O(\ln m - \ln ε)}$ time. We discuss numerical experiments and observe that for the random graph $G(n, 1/2)$ one can afford a much larger $γ$, provided the ratio $n/m$ is sufficiently large.

math.CO↗

arXiv2018

Approximating real-rooted and stable polynomials, with combinatorial applications

Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -δ$ for some $0<δ<1$. We show that for any $0 < ε<1$, the value of $p(1)$ is determined within relative error $ε$ by the coefficients $a_k$ with $k \leq {c \over \sqrtδ} \ln {n \over ε\sqrt{ δ}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < ε< 1$, the total number $M(G)=m_0(G)+m_1(G) + \ldots $ of matchings is determined within relative error $ε$ by the numbers $m_k(G)$ with $k \leq c \sqrtΔ \ln (v /ε)$, where $Δ$ is the largest degree of a vertex, $v$ is the number of vertices of $G$ and $c >0$ is an absolute constant. We prove a similar result for polynomials with complex roots satisfying $\Re\thinspace z \leq -δ$ and apply it to estimate the number of unbranched subgraphs of $G$.

math.CO↗

arXiv2017

Approximating permanents and hafnians

We prove that the logarithm of the permanent of an nxn real matrix A and the logarithm of the hafnian of a 2nx2n real symmetric matrix A can be approximated within an additive error 1 > epsilon > 0 by a polynomial p in the entries of A of degree O(ln n - ln epsilon) provided the entries a_ij of A satisfy delta < a_ij < 1 for an arbitrarily small delta > 0, fixed in advance. Moreover, the polynomial p can be computed in n^{O(ln n - ln epsilon)} time. We also improve bounds for approximating ln per A, ln haf A and logarithms of multi-dimensional permanents for complex matrices and tensors A.

math.CO↗