arXiv subjects

Paul Federbush

Publications and source records attributed to Paul Federbush.

At least 19 recordsLinked to original sources

arXiv2024

The Aesthetic Asymptotics of the Mayer Series Coefficients for a Dimer Gas on a Regular Lattice

We conjecture that for all regular lattices b(n) is asymptotically of the form in eq.(A1). (-1)^{n+1} b(n) = exp( k(-1) n + k(0) ln(n) + k(1) / n + k(2) / n^(2)...) (A1) We restrict testing this to lattices for which we know the first 20 Mayer series coefficients, the b(n). This includes the infinite number of rectangular lattices, one for each dimension, the tetrahedral lattice ( in this one case we know only the first 19 coefficients ), and the (bipartite) body centered cubic lattices, in dimensions 3 through 7. In this paper we will detail results for the rectangular lattices in dimensions 2,3,5,11,and 20, for the tetrahedral lattice, and for the body centered cubic lattices in dimensions 3,4, and 5. These are all bipartite, unfortunately we do not have an example of a non-bipartite regular lattice for which we know enough of the b(n) to work with. For the triangular lattice, regular and non-bipartite, we know the first 14 b(n). We feel this is not enough terms to make any judgement, hopefully someone may compute more terms. We work with an 'approximation' that keeps the first four terms, in k{-1), k(0), k(1), k(2), in the exponent in eq.(A1). Agreement will be striking. At the end of Part 1 there is a digression on a conjecture in line with recent applications of the renormalization group to study phase transitions and the ideas of Cardy, [10]. In Part 7 there is some study of susceptibility series for the Ising model on the 2d rectangular lattice, triangular lattice, and honeycomb lattice; where there is surprising similarity to Mayer series on regular graphs, as studied herein. Also in Part 7 we show, mirabile dictu, that the number of partitions function. p(n), has the 'magic' property. As does the ith prime function, actually ithprime(n-1).

math-ph↗

arXiv2023

A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0-1 Matrices

We consider two ensembles of nxn matrices. The first is the set of all nxn matrices with entries zeroes and ones such that all column sums and all row sums equal r, uniformly weighted. The second is the set of nxn matrices with zero and one entries where the probability that any given entry is one is r/n, the probabilities of the set of individual entries being i.i.d.'s. Calling the two expectation values E and EB respectively, we develop a formal relation E(perm(A)) = EB(perm (A)) x exp{sum Ti}. We also use a well-known approximating ensemble to E, E1. We prove using E or E1 one obtains the same value of Ti for i < 21. (THE PUBLISHED VERSION OF THIS PAPER ONLY OBTAINS RESULTS FOR i < 8. We go beyond the results of the published version by taking much more advantage of recent work of Pernici and of Wanless on i-matchings on regular bipartite graphs.)These terms Ti, i < 21, have amazing properties. We conjecture that these properties hold also for all i.

math-ph↗

arXiv2022

Four Amazing Positivities with Dimers/i-Matchings

We collect a number of striking recent results in a study of dimers on infinite regular bipartite lattices and also on regular bipartite graphs. We clearly separate rigorously proven results from conjectures. A primary goal is to show people: here is a field which is ripe for further interesting research. We separate four classes of endeavor, of which we here extract two items to whet one's appetite. Primo,for hyper-rectangular lattices of every dimension the first 20 virial coefficients are positive. (One has no understanding of this yet!) Secondo, all regular bipartite graphs with less than $14$ vertices satisfy graph positivity, defined below. (Here there is some understanding.)

math-ph↗

arXiv2022

Random Regular Bipartite Graphs Satisfy Weak Virial Positivity, for a Large Range of the Parameters

We deal with $r$-regular bipartite graphs with $2n$ vertices. In a previous paper, Butera, Pernici and the author have introduced a quantity $u(i)$, $u(i) = -\ln(i!m\_i)$, a function of the number of $i$-matchings, $m\_i$, and conjectured that the fraction of graphs that violate $Δ^k u(i) > 0$ for $k > 1$ vanishes as $n$ goes to infinity. Here $Δ$ is the finite difference operator. We now more particularly define the "Virial Positivity Conjecture" as the conjecture that the fraction of graphs that satisfy $Δ^k u(i)$ go to 0 for all $k > 1$ and $i$, approaches 1 as $n$ goes to infinity. The "Weak Virial Positivity Conjecture" is the conjecture that for each $i$ and $k > 1$ the probability that $Δ^k u(i) > 0$ goes to $1$ as $n$ goes to infinity. The term Virial is used since the condition $Δ^k u(i) > 0$ corresponds to the positivity of the Virial coefficients for infinite regular lattices. Herein we prove Weak Virial Positivity for the range of parameters $r < 11$, $i+k < 101$, $1 < k < 28$,or $i+k < 30$ all $r$. A formalism of Wanless as systematized by Pernici is central to this effort. Basically this paper is a corollary to our parallel attack on graph positivity in a previous paper. We assume basic knowledge of this previous paper.

math.CO↗

arXiv2022

A PROOF of Weak Graph Positivity, for a Large Range of the Parameters

One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the fraction of graphs that satisfy Delta^k( d(i)) is non-negative, for all k and i, approaches 1. Here Delta is the finite difference operator. This conjecture we called the "graph positivity conjecture". "Weak graph positivity" is the conjecture that for each i and k the probability that Delta^k (d(i) is non-negative goes to 1 as n goes to infinity. Here we prove this for the range of parameters where r < 11, i+k < 101, k < 21, or i+k < 30 all r. A formalism of Wanless as systematized by Pernici is central to this effort.

math.CO↗

arXiv2022

On the Pernici-Wanless Expansion for the Entropy ( and Virial Coefficients ) of a Dimer Gas on an Infinite Regular Lattice

We work with the following expression for the entropy (density) of a dimer gas on an infinite r-regular lattice lambda(p) = 1/2 [ pln(r)-ln(p)-2(1-p)ln(1-p)-p ]+sum_{k=2}(d_k)(p^k) where the indicated sum converges for density, p, small enough. Pernici has computed the coefficients d_k for k < 13. He found these d_k to be polynomials in certain interesting "geometric quantites" arising in the work of Wanless. Each of these quantities is the number density of isomorphic mappings of some graph into the lattice (graph). So for a bipartite lattice d_2 = c_2 d_3 = c_3 d_4 = c_4 + c_5 hat{G}_1 d_5 = c_6 + c_7 hat{G}_1. The c_i depend only on r. Here hat{G}_1 is the density of mapping classes of the four loop graph into the lattice. The limit of 1/V times the number of such mapping classes into a lattice of volume V as V goes to infinity. The infinite volume limit. There is a simple linear relation that yields the kth virial coefficient from the value of d_k! We feel this expression gives the deepest insight into the virial coefficients so far obtained. What we show in this paper is that such polynomial relations for the d_k in these geometric quantities holds for the d_k for k < 28. Of course we expect it to hold for all k. We use the same computation procedure as Pernici. We note this procedure is not rigorously established. So far a procedure for the physicist, perhaps not the mathematician (their loss). It is a worthy challenge for the mathematical physicist to supply the needed rigor.

math-ph↗

arXiv2020

The Genius Conjectures (via Bell Polynomials)

We present two related conjectures, arising in work on i-matchings in random r-regular bipartite graphs. The conjectures themselves are easily stated and involve only basic properties of convergent power series. One formulation involves Bell's polynomials. The conjectures name was chosen since we earnestly believe only a truly genius mathematician will prove them. We advise others not to try. A further belief is that the proof will arise from some deep properties of partitions.

math.CO↗

arXiv2019

A Near Proof of Weak Graph Positivity, A New Property of Random Regular Graphs

One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the fraction of graphs that satisfy Delta^k d(i) for all k and i, approaches 1. Here Delta is the finite difference operator. This conjecture we called the 'graph positivity conjecture'. In this paper it is formally shown that for each i and k the probability that Delta^k d(i) goes to 1 with n going to infinity. We call this weaker result the 'weak graph positivity conjecture ( theorem )'. A formalism of Wanless as systematized by Pernici is central to this effort. Our result falls short of being a rigorous proof since we make a sweeping conjecture ( computer tested ), of which we so far have only a portion of the proof.

math.CO↗

arXiv2018

A Set of Conjectured Identities for Stirling Numbers of the First Kind

Given an integer g, g > 1, an integer w, -1 < w <g - 1, and a set of g distinct numbers, c_1, ..., c_g, we present a conjectured identity for Stirling numbers of the first kind. We have proven all the equalities in case g < 7; and for the case g = 7, provided w < 4. These expressions arise from an aspect of the study of the dimer-monomer problem on regular graphs.

math.CO↗

arXiv2018

Proof of Convergence for the Lattice Monomer-Dimer Cluster Expansion I, a Simplified Model

We present some promising ideas to treat the problem of making completely rigorous the development of our expression for $λ_d(p)$ of the monomer-dimer problem on a $d$-dimensional hypercubic lattice \begin{equation}\label{abstract1} λ_d(p)=\frac{1}{2}\Big(p\ln(2d)-p\ln(p)-2(1-p)\ln(1-p)-p\Big) +\sum_{k=2}a_k(d)p^k \end{equation} where $a_k(d)$ is a sum of powers $(1/d)^r$ for \begin{equation}\label{abstract2} k-1\leq r\leq k/2 \end{equation} In fact as we will point out one has allready rigorously established the convergence of the sum in expression for $λ$ for small $p$. It is the $d$ dependence of $a_k(d)$ that has yet to be rigorously shown. We do not now know how to complete the proof.

math-ph↗

arXiv2018

Regular Bipartite Lattices with Large Values of Theta_2,2,2/C_4

The quantities C_4 and Theta_2,2,2 are as defined by Wanless, C_4 just the number of 4-loops of a graph. The construction of this paper provides a counterexample to a conjecture of Butera, Pernici, and the author about the monomer-dimer entropy, lambda, of a regular bipartite lattice. The lattice we construct is not a lattice graph in its most common definition.

math.CO↗

arXiv2017

On the Approximate Asymptotic Statistical Independence of the Permanents of 0-1 Matrices

We consider the ensemble of n x n 0 - 1 matrices with all column and row sums equal r. We give this ensemble the uniform weighting to construct a measure E. We know from the work of Wanless and Pernici that E(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E(perm_{m_i}(A)) * (1+ O(1/n^4)) In this paper we prove E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^2)) where E_1 is the measure constructed on the ensemble of n x n 0 - 1 matrices with non-negative integer entries realized as the sum of r random permutation matrices. E_1 is often used as an "approximation" to E. We have computer evidence for E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^4)).

math.CO↗

arXiv2016

Asymptotic Behavior of the Expectation Value of Permanent Products, a Sequel

Continuing the computations of the previous paper,[1], we calculate another approximation to the expectation value of the product of two permanents in the ensemble of 0-1 n x n matrices with like row and column sums equal r uniformly weighted. Here we consider the Bernoulli random matrix ensemble where each entry independently has a probability p=r/n of being one, otherwise zero. We denote the expectations of the approximation ensemble of [1] by E, and the expectations of the present approximation ensemble, the Bernoulli random matrix ensemble, by E*. One has for these lim_{r to infinity}( lim_{n to infinity} (1/n) ln(E(perm_m(A))) -lim_{n to infinity} (1/n) ln(E*(perm_m(A))) ) = 0 and lim_{n to infinity} (1/n) ln(E(perm_m(A)perm_m'(A))) = lim_{n to infinity} (1/n) ln(E(perm_m(A))) + lim_{n to infinity} (1/n) ln(E(perm_m'(A))) Here and in all such formulas the subscripts m,m' are assumed proportional to n. It seems likely to us that lim_{r to infinity}( lim_{n to infinity} (1/n) ln(E*(perm_m(A)perm_m'(A))) - lim_{n to infinity} (1/n) ln(E*(perm_m(A))) + - lim_{n to infinity} (1/n) ln(E*(perm_m'(A))) ) = 0 We believe: "E gives us the `correct' expectations in these equations, and E* is only `correct' in the r to infinity limit."

math.CO↗

arXiv2014

Asymptotic Behavior of the Expectation Value of Permanent Products

We would desire to have done the calculations of this paper in the measure on nxn matrices that weights uniformly all 0-1 matrices with row and column sum equal to r, other matrices given weight zero. Instead we work with all matrices that are the sum of r independent uniformly weighted permutation matrices, with the hope that the computations we perform give the same result in this measure. We derive the result for limiting expectations lim (1/n)ln(E(perm_m(A) perm_m'(A))) =lim (1/n)ln(E(perm_m(A)))+ +lim (1/n)ln(E(perm_m'(A))) Here the limit is n to infinity, r is fixed, and m and m' are taken as each proportional to n.

math.CO↗

arXiv2014

Existence of Magnetic Compressible Fluid Stars

The existence of magnetic star solutions which are axi-symmetric stationary solutions for the Euler-Poisson system of compressible fluids coupled to a magnetic field is proved in this paper by a variational method. Our method of proof consists of deriving an elliptic equation for the magnetic potential in cylindrical coordinates in $\mathbb{R}^3$, and obtaining the estimates of the Green's function for this elliptic equation by transforming it to 5-Laplacian.

math.AP↗

arXiv2013

Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices

Recently an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise an expansion for the entropy, dependent on the dimer-density p, of a monomer-dimer system, involving a sum sum_k a_k(d) p^k, has been recently offered. We herein extend the number of the known expansion coefficients from 6 to 20 for the hyper-cubic lattices of general dimension d and from 6 to 24 for the hyper-cubic lattices of dimensions d < 5 . We show that this extension can lead to accurate numerical estimates of the p-dependent entropy for lattices with dimension d > 2. The computations of this paper have led us to make the following marvelous conjecture: "In the case of the hyper-cubic lattices, all the expansion coefficients, a_k(d), are positive"! This paper results from a simple melding of two disparate research programs: one computing to high orders the Mayer series coefficients of a dimer gas, the other studying the development of entropy from these coefficients. An effort is made to make this paper self-contained by including a review of the earlier works.

hep-lat↗

arXiv2013

For the Monomer-Dimer lambda_d(p), the Master Algebraic Conjecture

The author has recently presented two different expressions for lambda_d(p) of the monomer-dimer problem involving a power series in p, the first jointly with Shmuel Friedland. These two expressions are certainly equal, but this has not yet been proven rigorously. The first is naturally developed from quantities J_i, cluster expansion kernels. The second from the Mayer (or Virial) series of a dimer gas, in particular from the b_i coefficients in the Mayer series. The sets {b_i} and {J_i} can be derived from each other. Given an arbitrary set of values for either the b_i or the J_i, both expressions may be given in terms of a formal sum. The master algebraic conjecture is that these two expressions are equivalent. This is detailed in the special case all J_i are zero.

math-ph↗

arXiv2012

The Dimer Gas Mayer Series, the Monomer-Dimer lambda_d(p), the Federbush Relation

The author and Shmuel Friedland recently presented an expression for lambda_d(p) of the monomer-dimer problem involving a power series in p . Herein I present a simple way to derive this expression for lambda_d(p) from the Mayer (or Virial) series of a dimer gas. The derivation is an exercise in basic statistical mechanics. We expect the relationship to be very useful.

math-ph↗