From 8f04b9ec8ac6d530e5044fe42051f658e6f251b0 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 22 Oct 2018 21:00:26 -0400
Subject: added referee response

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+%
+%  test.tex - description
+%
+%  Created by  on Mon Oct 22 19:11:00 EDT 2018.
+%  Copyright (c) 2018 pants productions. All rights reserved.
+%
+\documentclass[fleqn]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,fullpage}
+
+\mathtoolsset{showonlyrefs=true}
+
+\title{LS16406 Referee Response}
+\author{Jaron Kent-Dobias \& James P Sethna}
+
+\begin{document}
+
+\def\[{\begin{equation}}
+\def\]{\end{equation}}
+
+\maketitle
+
+We address the referee response to our first submission below.
+
+\begin{quote}
+\begin{verbatim}
+As mentioned before, other methods for carrying out cluster
+simulations in a field have been proposed. See:
+
+V. Martin-Mayor, D. Yllanes Phys. Rev. E 80 (2009), 015701
+
+In this paper it is shown how to construct a cluster algorithm with an
+arbitrary conserved physical quantity, by working in a different
+statistical ensemble. This is equivalent to fixing the conjugated
+field: both are related by a Legendre transformation, dominated by a
+saddle point, so there is a one-to-one relationship (fixing a
+magnetization at a value m = c is equivalent to fixing the magnetic
+field at a value h* such that <m>_h* = c). The method, Tethered Monte
+Carlo, also works in situations where more than one order parameter is
+fixed (or more than one field is applied): V. Martin-Mayor, B. Seoane,
+D. Yllanes, J. Stat. Phys. 144 (2011) 554.
+
+Of course, the present approach is very different from that of the
+above reference, but perhaps the authors could address the
+differences.
+\end{verbatim}
+\end{quote}
+
+The indicated paper is indeed interesting, and indeed different in the ways
+already outlined by the referee: the algorithm described operates in a
+different statistical ensemble. Extracting values in the constant-field
+ensemble is done by a numeric integral over results from simulations of many
+constant-magnetization systems. Another difference is that the algorithm
+described relies on a heat-bath method to update the clusters once formed, and
+therefore spiritually belongs to a broad class of existing algorithms we
+already cite that form clusters and decide to flip them using metropolis or
+heat-bath methods directly in the constant field ensemble. Reference to this
+work has now been made alongside these others. Notably, a constant-magnetization ensemble
+cluster algorithm that uses clusters without the need for a separate auxiliary
+update exists; see JR Heringa \& HWJ Blote, Phys Rev E 57 5 (1998), 4976. This
+latter
+work feels like a nearer analogue to our own.
+
+\begin{quote}
+\begin{verbatim}
+The part of the paper dealing with numerical tests of the method is
+severely lacking in detail. First of all, the authors just say, after
+eq. (12), that they measure tau "with standard methods", but cite a
+paper from 1992 with a different approach to what is commonly done
+nowadays. A useful reference could be
+
+G. Ossola, A.D. Sokal, "Dynamic critical behavior of the Swendsen–Wang
+algorithm for the three-dimensional Ising model" Nucl. Phys. B 691
+(2004) 259, https://doi.org/10.1016/j.nuclphysb.2004.04.026
+\end{verbatim}
+\end{quote}
+
+We found the suggested reference very helpful, and now use the methods
+described therein for computation of correlation times and their uncertainties.
+
+\begin{quote}
+\begin{verbatim}
+In any case, more detailed is needed on the computation of tau, such
+as showing some autocorrelation functions and explaining how the error
+bars are estimated (this could be an appendix).
+\end{verbatim}
+\end{quote}
+
+Since the autocorrelation times and their uncertainties are now computed using
+the method suggested above, explicit reference to that method seems sufficient
+to explain how the data shown were processed. The autocorrelation functions
+themselves appear unremarkable pure exponentials, as the energy
+autocorrelation functions also were found to be in Ossola \& Sokal. Moreover,
+we compute $\tau$ for six models at at least seven system sizes and at least
+fifteen values of the field, meaning that there are hundreds of independent
+autocorrelation functions, perhaps beyond the scope of even an appendix.
+
+\begin{quote}
+\begin{verbatim}
+A direct computation of z with their data would be much preferable
+to the scaling collapses, which are semi-quantitative. Why has this
+not been attempted?
+\end{verbatim}
+\end{quote}
+
+In the revised manuscript we provide rough estimates for $z$ in the models
+studied, but reiterate that since the algorithm is identical to Wolff for
+trivial fields, $z$ is simply that of the Wolff algorithm on each model.  We
+are principally interested in exploring the way the autocorrelation time
+scales as one moves away from the zero field critical point---where the
+dynamic behavior of the algorithm is already known---in the nonzero field
+direction. Remeasuring $z$ for the Wolff algorithm does not accomplish this;
+we believe that showing the scaling collapses, which in turn outline the form
+of underlying universal scaling functions, does.
+
+\begin{quote}
+\begin{verbatim}
+As another general point, the authors should provide some technical
+details of their simulations, such as the number of MC steps. For
+systems other than the 2D Ising model not even the sizes simulated are
+specified.
+\end{verbatim}
+\end{quote}
+
+Information about system sizes has been added.  Since the work involves so
+many separate data points, including such details for each would greatly
+increase the size of the manuscript without adding much useful information. At
+least $10^6$ runs were preformed for every data point involving
+autocorrelation times.
+
+\begin{quote}
+\begin{verbatim}
+In Fig. 1, the authors show results for the 2D Ising model up to
+sizes L = 256. This is a very small size for such a simple system,
+especially considering that the point of these cluster algorithms is
+that there is no critical slowing down. The figure should include a
+legend saying which curve corresponds to which system size.
+\end{verbatim}
+\end{quote}
+
+A $512\times512$ curve has been added, along with system size labels. We
+emphasize that unlike Ossola \& Sokal, this is not meant to be a precision study of any of these
+models, for which extensive computer time might be dedicated to measuring
+quantities for much larger systems, as we ourselves have done in another preprint
+(arXiv:1707.03791 [cond-mat.stat-mech]). We believe the behavior we intend to
+show---the way the algorithm scales as the critical point is departed in the
+field direction---is demonstrated well by the system sizes used.
+
+\begin{quote}
+\begin{verbatim}
+Why is tau only computed for the Ising model? In Fig. 2 the
+efficiency of the method is demonstrated via a more indirect method
+for the other systems. In addition, this figure does not even say
+which system sizes have been simulated.
+\end{verbatim}
+\end{quote}
+
+Autocorrelation times have been computed for the other models studied. System
+size labels have been added to all figures.
+
+\begin{quote}
+\begin{verbatim}
+As the authors say, "the goal of statistical mechanics is to compute
+expectation values of observables". In this sense, why don't the
+authors compute some simple physical observable, such as the energy,
+and show how much precision can be achieved for a given computational
+effort? At the end of the day, this is the true measure of efficiency
+for any Monte Carlo method.
+\end{verbatim}
+\end{quote}
+
+A great deal is known about the efficiency of the Wolff algorithm in this
+regard. The algorithm described here is exactly the same as the Wolff
+algorithm when there is no coupling to an external field. We hope that our
+numeric experiments convincingly demonstrate that this algorithm's efficiency
+scales from the already known zero field Wolff behavior into nonzero field as
+an ordinary scaling analysis would predict. The supplied autocorrelation times
+are already an indication of how much precision can be achieved for a given
+computational effort.
+
+\end{document}
+
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