Search

We solve analytically the differential equations for a skier on a circular hill and for a particle on a loop-the-loop track when the hill or track is endowed with a coefficient of kinetic friction $\mu$. For each problem, we determine the exact "phase diagram" in the two-dimensional parameter plane., Comment: LaTeX2e, 22 pages (main text plus two appendices) including 10 figures. Version 2 is much revised, thanks to helpful comments by referees. To be published in the American Journal of Physics

We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\mathbf{\phi})$ in infinitely many indeterminates that give a weight $y$ to each improper edge and a weight $m! \, \phi_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\mathbf{\phi}$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays., Comment: LaTeX2e, 75 pages, includes 16 figures. arXiv admin note: substantial text overlap with arXiv:2105.05583

We show that very simple continued fractions can be obtained for the ordinary generating functions enumerating permutations or D-permutations with a large number of independent statistics, when each cycle is given a weight $-1$. The proof is based on a simple lemma relating the number of cycles modulo 2 to the numbers of fixed points, cycle peaks (or cycle valleys), and crossings., Comment: LaTeX2e, 25 pages, includes 4 figures. arXiv admin note: text overlap with arXiv:2304.06545, arXiv:2212.07232, arXiv:2003.08192, arXiv:2304.14487

I show that a hypergeometric function ${}_{p}F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;\,\cdot\,)$ with $p \le q$ belongs to the Laguerre--Pólya class $LP^+$ for arbitrarily large $b_{p+1},\ldots,b_q > 0$ if and only if, after a possible reordering, the differences $a_i - b_i$ are nonnegative integers. This result arises as an easy corollary of the case $p=q$ proven two decades ago by Ki and Kim. I also give explicit examples for the case ${}_{1}F_2$., 10 pages, LaTeX2e

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them., Comment: LaTeX2e, 122 pages, includes 9 tikz figures. Version 2 includes a new footnote 27 on page 81. To be published in Advances in Applied Mathematics

I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$., Comment: LaTeX2e, 11 pages. Version 2 fixes Remark 2 on page 5. To be published in Proceedings of the AMS

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of $T(a,c,0,e,0,0)$, which includes the reversed Stirling subset triangle., Comment: LaTeX2e, 12 pages with 2 figures: extended abstract submitted to FPSAC 2021

This is a response to five comments [American Psychologist 69, 626-629 and 632-635 (2014)] on our article arXiv:1307.7006., Comment: PDF, 9 pages

We argue that the 4-state Potts antiferromagnet has a finite-temperature phase transition on any Eulerian plane triangulation in which one sublattice consists of vertices of degree 4. We furthermore predict the universality class of this transition. We then present transfer-matrix and Monte Carlo data confirming these predictions for the cases of the union-jack and bisected hexagonal lattices., Comment: LaTeX2e/Revtex4, 5 pages

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z \ge \alpha/\nu is close to but probably not sharp in d=2, and is far from sharp in d=3, for all q. The conjecture z \ge \beta/\nu is false (for some values of q) in both d=2 and d=3., Comment: Revtex4, 4 pages including 4 figures

I present and analyze a quadratically convergent algorithm for computing the infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n) = \sum_{m=0}^\infty {(-t)^m x^{m(m+1)/2} \over (1-x)(1-x^2) ... (1-x^m)} due to Euler. The efficiency of the algorithm deteriorates as |x| \uparrow 1, but much more slowly than in previous algorithms. The key lemma is a two-sided bound on the Dedekind eta function at pure imaginary argument, ��(iy), that is sharp at the two endpoints y=0,\infty and is accurate to within 9.1% over the entire interval 0 < y < \infty.

Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x^\star = \lim_{L\to\infty} \xi(L)/L and the first four magnetization moment ratios V_{2n} = /^n. As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G_{2n}^*. We confirm these predictions by a high-precision Monte Carlo simulation., Comment: 36 pages (LaTeX with style files included). Paper drastically shortened: discussion of corrections to scaling and SW dynamics deferred to future work. Analysis of higher renormalized coupling constants G_{2n}^* added. New bibliography added. To appear in J. Stat. Phys

We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an $XY$ model into the target model, and we study its mathematical properties for a variety of nonlinear $��$-models. We then apply the method to the two-dimensional $O(N)$-symmetric nonlinear $��$-models (also called $N$-vector models) with $N=3,4,8$ and study its dynamic critical behavior. Using lattices up to $256 \times 256$, we find dynamic critical exponents $z_{int,{\cal M}^2} \approx 0.70 \pm 0.08$, $0.60 \pm 0.07$, $0.52 \pm 0.10$ for $N=3,4,8$, respectively (subjective 68\% confidence intervals). Thus, for these asymptotically free models, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with $N$. We also analyze the static data for $N=8$ using a finite-size-scaling extrapolation method. The correlation length $��$ agrees with the four-loop asymptotic-freedom prediction to within $\approx 1\%$ over the interval $12 < ��< 650$., 75 pages (including 15 figures); 610216 bytes uuencoded gzip'ed (expands to latex file, 24 postscript figures and 3 style files)

We reply to criticism by Patrascioiu and Seiler [hep-lat/9502019] of our results [Phys. Rev. Lett. 75, 1891 (1995), hep-lat/9411009] on asymptotic scaling in the two-dimensional $O(3)$ $\sigma$-model, which were based on a finite-size-scaling extrapolation method., Comment: Postscript file avalaible at http://www1.le.infn.it:8080/~caraccio/96.html

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $��$ up to $\approx\! 4 \times 10^5$, using a multi-grid Monte Carlo (MGMC) algorithm. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. For $��\gtapprox 10^3$ we observe good asymptotic scaling in the bare coupling; at $��\approx 4 \times 10^5$ the nonperturbative constant is within 2--3\% of its predicted limiting value., 13 pages (includes 3 figures), self-unpacking uuencoded .tar.gz

We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen-Wang algorithm for the two-dimensional 3-state Potts model. We find that the Li-Sokal bound ($\tau_{int,E} \geq const \times C_H$) is almost but not quite sharp. The ratio $\tau_{int,E} / C_H$ seems to diverge either as a small power ($\approx 0.08$) or as a logarithm., Comment: 35 pages including 3 figures. Self-unpacking file containing the LaTeX file, the needed macros (epsf.sty, indent.sty, subeqnarray.sty, and eqsection.sty) and the 3 Postscript figures. Revised version fixes a normalization error in \xi (with many thanks to Wolfhard Janke for finding the error!). To be published in J. Stat. Phys. 87, no. 1/2 (April 1997)

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation and a W-cycle, applied to the one-dimensional $O(4)$-symmetric nonlinear $��$-model [= $SU(2)$ principal chiral model], on lattices from $L=128$ to $L=16384$. Our data for the integrated autocorrelation time $��_{int,{\cal M}^2}$ are well fit by a logarithmic growth. We have no idea why the critical slowing-down is not completely eliminated., 377866 bytes Postscript, 16 pages, includes figures

We introduce a new and very convenient approach to multi-grid Monte Carlo (MGMC) algorithms for general nonlinear $��$-models: it is based on embedding an $XY$ model into the given $��$-model, and then updating the induced $XY$ model using a standard $XY$-model MGMC code. We study the dynamic critical behavior of this algorithm for the two-dimensional $O(N)$ $��$-models with $N = 3,4,8$ and for the $SU(3)$ principal chiral model. We find that the dynamic critical exponent $z$ varies systematically between these different asymptotically free models: it is approximately 0.70 for $O(3)$, 0.60 for $O(4)$, 0.50 for $O(8)$, and 0.45 for $SU(3)$. It goes without saying that we have no theoretical explanation of this behavior., 75280 bytes uuencoded gzip'ed (expands to 185401 bytes Postscript); 4 pages including all figures; contribution to Lattice '95

In this talk we present the exact solution of the most general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbour interactions, and we discuss the possible continuum limits. All these results are obtained using a high-temperature expansion in terms of hyperspherical harmonics. Applications in higher dimensions of the same technique are then discussed., 59208 bytes uuencoded gzip'ed (expands to 135067 bytes Postscript); 4 pages including all figures; contribution to Lattice '95

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $��$-model at correlation lengths $��$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $��\sim 10^2$, decreases to $\approx 4\%$ at $��\sim 10^5$., 11 pages including 3 figures, 278964 bytes Postscript (changed storage format)

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $��$-model at correlation lengths $��$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $��\sim 10^2$, decreases to $\approx 4\%$ at $��\sim 10^5$., 3 pages including 3 figures, 170894 bytes Postscript, contribution to Lattice '94; see also hep-lat/9409004 and hep-lat/9411009

We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\infty)$ $��$-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice., 11 pages including 4 figures, 280216 bytes Postscript (NYU-TH-94/09/02)

We argue that the two-dimensional $O(N)$-invariant lattice $��$-model with mixed isovector/isotensor action has a one-parameter family of nontrivial continuum limits, only one of which is the continuum $��$-model constructed by conventional perturbation theory. We test the proposed scenario with a high-precision Monte Carlo simulation for $N=3,4$ on lattices up to $512 \times 512$, using a Wolff-type embedding algorithm. [CPU time $\approx$ 7 years IBM RS-6000/320H] The finite-size-scaling data confirm the existence of the predicted new family of continuum limits. In particular, the $RP^{N-1}$ and $N$-vector models do not lie in the same universality class., 10 pages (includes 2 figures), 211176 bytes Postscript, NYU-TH-93/07/03, IFUP-TH 34/93

We study a class of Monte Carlo algorithms for the nonlinear $��$-model, based on a Wolff-type embedding of Ising spins into the target manifold $M$. We argue heuristically that, at least for an asymptotically free model, such an algorithm can have dynamic critical exponent $z \ll 2$ only if the embedding is based on an (involutive) isometry of $M$ whose fixed-point manifold has codimension 1. Such an isometry exists only if the manifold is a discrete quotient of a product of spheres. Numerical simulations of the idealized codimension-2 algorithm for the two-dimensional $O(4)$-symmetric $��$-model yield $z_{int,{\cal M}^2} = 1.5 \pm 0.5$ (subjective 68\% confidence interval), in agreement with our heuristic argument., 70 pages, 7 postscript figures

We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC) algorithm with piecewise-constant interpolation applied to the two-dimensional O(4)-symmetric nonlinear $��$-model [= SU(2) principal chiral model], on lattices up to $256 \times 256$. We find a dynamic critical exponent $z_{int,{\cal M}^2} = 0.60 \pm 0.07$ for the W-cycle and $z_{int,{\cal M}^2} = 1.13 \pm 0.11$ for the V-cycle, compared to $z_{int,{\cal M}^2} = 2.0 \pm 0.15$ for the single-site heat-bath algorithm (subjective 68% confidence intervals). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. For a $256 \times 256$ lattice, W-cycle MGMC is about 35 times as efficient as a single-site heat-bath algorithm.

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_G(q) for m \times n rectangular subsets of the square lattice, with m \le 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n \to\infty, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B_2,B_3,B_4,B_5 are limiting points of partition-function zeros as n \to\infty whenever the strip width m is \ge 7 (periodic transverse b.c.) or \ge 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B_{10}) cannot be a chromatic root of any graph. Comment: 90 pages, LaTeX2e. Self-unpacking archive containing the tex file, three sty files, 39 ps files, and a Mathematica file (transfer1.m). Many small changes to improve clarity and new Section 7.3 added. To be published in J. Stat. Phys

We study the chromatic polynomial P_G(q) for m \times n triangular-lattice strips of widths m, 105 pages, LaTeX2e. Self-unpacking archive containing the tex file, three sty files, 36 ps files, and a Mathematica file (transfer3.m). The background material can be found in cond-mat/0004330 and cond-mat/0011456. Section 6 has been rewritten and new results are presented. To be published in J. Stat. Phys

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $\xi$ up to $\sim 4 \times 10^5$, using a multi-grid Monte Carlo (MGMC) algorithm and approximately one year of Cray C-90 CPU time. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 12%$ at $\xi \sim 25$, decreases to $\approx 2%$ at $\xi \sim 4 \times 10^5$. We also analyze the dynamic critical behavior of the MGMC algorithm using lattices up to $256 \times 256$, finding the dynamic critical exponent $z_{int,{\cal M}^2} \approx 0.45 \pm 0.02$ (subjective 68% confidence interval). Thus, for this asymptotically free model, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated. Comment: self-unpacking archive including .tex, .sty and .ps files; 126 pages including all figures