arXiv subjects

Mohan Ravichandran

Publications and source records attributed to Mohan Ravichandran.

17 recordsLinked to original sources

arXiv2025

The bunkbed problem and the random cluster model

The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of the class of random cluster measures. We show that the major partial (positive) results on the bunkbed conjecture can also be proved for all random cluster measures, including the results for complete graphs, complete bipartite graphs, and the case when $p \uparrow 1$. The arboreal gas measure for forests is another limit of the random cluster measure for which we conjecture the inequality to be true and provide proofs in special cases. We identify a setting where the conjecture does hold, that of ``almost spanning tree measures''. A further analysis leads to intriguing correlation inequalities that complement Rayleigh's inequalities for spanning tree measures.

math.PR↗

arXiv2023

Quasipolynomial-time algorithms for Gibbs point processes

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a finite-range stable potential. This result holds for all activities $λ$ for which the partition function satisfies a zero-free assumption in a neighborhood of the interval $[0,λ]$. As a corollary, for all finite-range stable potentials we obtain a quasipolynomial-time determinsitic algorithm for all $λ< /(e^{B + 1} \hat C_ϕ)$ where $\hat C_ϕ$ is a temperedness parameter and $B$ is the stability constant of $ϕ$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $λ< e/Δ_ϕ$, where $Δ_ϕ$ is the potential-weighted connective constant of the potential $ϕ$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.

cs.DS↗

arXiv2022

Deterministic Approximation Algorithms for Volumes of Spectrahedra

We give a method for computing asymptotic formulas and approximations for the volumes of spectrahedra, based on the maximum-entropy principle from statistical physics. The method gives an approximate volume formula based on a single convex optimization problem of minimizing $-\log \det P$ over the spectrahedron. Spectrahedra can be described as affine slices of the convex cone of positive semi-definite (PSD) matrices, and the method yields efficient deterministic approximation algorithms and asymptotic formulas whenever the number of affine constraints is sufficiently dominated by the dimension of the PSD cone. Our approach is inspired by the work of Barvinok and Hartigan who used an analogous framework for approximately computing volumes of polytopes. Spectrahedra, however, possess a remarkable feature not shared by polytopes, a new fact that we also prove: central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume. This allows for very general approximation algorithms, which apply to large classes of naturally occurring spectrahedra. We give two main applications of this method. First, we apply this method to what we call the "multi-way Birkhoff spectrahedron" and obtain an explicit asymptotic formula for its volume. This spectrahedron is the set of quantum states with maximal entanglement (i.e., the quantum states having univariant quantum marginals equal to the identity matrix) and is the quantum analog of the multi-way Birkhoff polytope. Second, we apply this method to explicitly compute the asymptotic volume of central sections of the set of density matrices.

cs.CG↗

arXiv2022

Chainlink Polytopes and Ehrhart-Equivalence

We introduce a class of polytopes that we call chainlink polytopes and which allow us to construct infinite families of pairs of non isomorphic rational polytopes with the same Ehrhart quasi-polynomial. Our construction is based on circular fence posets, which admit a non-obvious and non-trivial symmetry in their rank sequences that turns out to be reflected in the polytope level. We introduce the related class of chainlink posets and show that they exhibit the same symmetry properties. We further prove an outstanding conjecture on the unimodality of circular rank polynomials.

math.CO↗

arXiv2021

Rank Polynomials of Fence Posets are Unimodal

We prove a conjecture of Morier-Genoud and Ovsienko that says that rank polynomials of the distributive lattices of lower ideals of fence posets are unimodal. We do this by introducing a related class of circular fence posets and proving a stronger version of the conjecture due to McConville, Sagan and Smyth. We show that the rank polynomials of circular fence posets are symmetric and conjecture that unimodality holds except in some particular cases. We also apply the recent work of Elizalde, Plante, Roby and Sagan on rowmotion on fences and show many of their homomesy results hold for the circular case as well.

math.CO↗

arXiv2021

Approximation algorithms for the random-field Ising model

Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree $Δ$. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and $Δ$. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size $Θ(\log n)$, are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree.

cs.DS↗

arXiv2021

Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity

We characterize all signed Minkowski sums that define generalized permutahedra, extending results of Ardila-Benedetti-Doker (2010). We use this characterization to give a complete classification of all positive, translation-invariant, symmetric Minkowski linear functionals on generalized permutahedra. We show that they form a simplicial cone and explicitly describe their generators. We apply our results to prove that the linear coefficients of Ehrhart polynomials of generalized permutahedra, which include matroid polytopes, are non-negative, verifying conjectures of De Loera-Haws-Koeppe (2009) and Castillo-Liu (2018) in this case. We also apply this technique to give an example of a solid-angle polynomial of a generalized permutahedron that has negative linear term and obtain inequalities for beta invariants of contractions of matroids.

math.CO↗

arXiv2018

Mixed Determinants and the Kadison-Singer problem

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison-Singer problem to prove improved paving estimates. Working with Anderson's paving formulation of Kadison-Singer instead of Weaver's vector balancing version, we show that the machinery of interlacing polyomials due to Marcus, Spielman and Srivastava works in this setting as well. The relevant expected characteristic polynomials turn out to be related to the so called "mixed determinants" that have been carefully studied by Borcea and Branden. This technique allows us to show that any projection with diagonal entries strictly less than $\frac{1}{4}$ can be two paved, matching recent results of Bownik, Casazza, Marcus and Speegle, though our estimates are asymptotically weaker. We also show that any projection with diagonal entries at most $\frac{1}{2}$ can be four paved, yielding improvements over currently known estimates. We also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have some remarkable combinatorial properties.

math.FA↗

arXiv2017

Asymptotically Optimal Multi-Paving

Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions $A^{(1)}, \cdots, A^{(k)} \in M_n(\mathbb{C})$ and $ε> 0$, one may find a paving $X_1 \amalg \cdots \amalg X_r = [n]$ where $r \leq 18kε^{-2}$ such that, \[λ_{max} (P_{X_i} A^{(j)} P_{X_i}) < ε, \quad i \in [r], \, j \in [k].\] As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be $(O(ε^{-2}),ε)$ paved. As an application, we give a simplified proof wth slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices.

math.FA↗

arXiv2017

An analytical Lieb-Sokal lemma

A polynomial $p \in \mathbb{R}[z_1, \cdots, z_n]$ is called real stable if it is non-vanishing whenever all the variables take values in the upper half plane. A well known result of Elliott Lieb and Alan Sokal states that if $p$ and $q$ are $n$ variate real stable polynomials, then the polynomial $q(\partial)p := q(\partial_1, \cdots, \partial_n)p$, is real stable as well. In this paper, we prove analytical estimates on the locations on the zeroes of the real stable polynomial $q(\partial)p$ in the case when both $p$ and $q$ are multiaffine, an important special case, owing to connections to negative dependance in discrete probability. As an application, we prove a general estimate on the expected characteristic polynomials upon sampling from Strongly Rayleigh distributions. We then use this to deduce results concerning two classes of polynomials, mixed characteristic polynomials and mixed determinantal polynomials, that are related to the Kadison-Singer problem.

math.FA↗

arXiv2016

Principal submatrices, restricted invertibility and a quantitative Gauss-Lucas theorem

We apply the techniques developed by Marcus, Spielman and Srivastava, working with principal submatrices in place of rank $1$ decompositions to give an alternate proof of their results on restricted invertibility. We show that one can find well conditioned column submatrices all the way upto the so called modified stable rank. All constructions are algorithmic. A byproduct of these results is an interesting quantitative version of the classical Gauss-Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq \frac{1}{2}$, the area of the convex hull of the roots of $p^{(cn)}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.

math.FA↗

arXiv2014

The Schur-Horn theorem in von Neumann algebras

A few years ago, Richard Kadison thoroughly analysed the diagonals of projection operators on Hilbert spaces and asked the following question: Let $\mathcal{A}$ be a masa in a type $II_1$ factor $\mathcal{M}$ and let $A \in \mathcal{A}$ be a positive contraction. Letting $E$ be the canonical normal conditional expectation from $\mathcal{M}$ to $\mathcal{A}$, can one find a projection $P \in \mathcal{M}$ so that [E(P) = A?] In a later paper, Kadison and Arveson, as an extension, conjectured a Schur-Horn theorem in type $II_1$ factors. In this paper, I give a proof of this conjecture of Arveson and Kadison. I also prove versions of the Schur-Horn theorem for type $II_{\infty}$ and type $III$ factors as well as finite von Neumann algebras.

math.OA↗

arXiv2014

Multivariable Schur-Horn theorems

We prove a variety of results describing the possible diagonals of tuples of commuting hermitian operators in type $II_1$ factors. These results are generalisations of the classical Schur-Horn theorem to the infinite dimensional, multivariable setting. Our description of these possible diagonals uses a natural generalisation of the classical notion of majorization to the multivariable setting. In the special case when both the given tuple and the desired diagonal have finite joint spectrum, our results are complete. When the tuples do not have finite joint spectrum, we are able to prove strong approximate results. Unlike the single variable case, the multivariable case presents several surprises and we point out obstructions to extending our complete description in the finite spectrum case to the general case. We also discuss the problem of characterizing diagonals of commuting tuples in $\mathcal{B}(\mathcal{H})$ and give approximate characterizations in this case as well.

math.OA↗

arXiv2011

The Schur-Horn theorem for operators with finite spectrum

The carpenter problem in the context of $II_1$ factors, formulated by Kadison asks: Let $\mathcal{A} \subset \mathcal{M}$ be a masa in a type $II_1$ factor and let $E$ be the normal conditional expectation from $\mathcal{M}$ onto $\mathcal{A}$. Then, is it true that for every positive contraction $A$ in $\mathcal{A}$, there is a projection $P$ in $\mathcal{M}$ such that $E(P) = A$? In this note, we show that this is true if $A$ has finite spectrum. We will then use this result to prove an exact Schur-Horn theorem for (positive)operators with finite spectrum and an approximate Schur-Horn theorem for general (positive)operators.

math.OA↗

arXiv2011

The Radial Masa in a Free Group Factor is Maximal Injective

The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\ \bar{\otimes}\ B_2$ is maximal injective in $M_1\ \bar{\otimes}\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors.

math.OA↗

arXiv2007

On Transitive Algebras Containing a Standard Finite von Neumann Subalgebra

Let $\M$ be a finite von Neumann algebra acting on a Hilbert space $\H$ and $Å$ be a transitive algebra containing $\M'$. In this paper we prove that if $Å$ is 2-fold transitive, then $Å$ is strongly dense in $\B(\H)$. This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [Ha1]) is 2-fold transitive, then $Å$ is strongly dense in $\B(\H)$. Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., $Ł{\mathbb{F}_n}$ and $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, are studied. Brown measures of certain operators in $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$ are explicitly computed.

math.OA↗

arXiv2006

A Folner Invariant for Type II_1 Factors

In this article we introduce an isomorphism invariant for type II_1 factors using the Connes-Folner condition. We compute bounds of this number for free group factors.

math.OA↗