arXiv subjects

K. Markström

Publications and source records attributed to K. Markström.

14 recordsLinked to original sources

arXiv2023

Revising the universality class of the four-dimensional Ising model

The aim of this paper is to determine the behavior of the specific heat of the 4-dimensional Ising model at the critical temperature, and via that determine if the Ising model and the $ϕ^4$-model belong to the same universality class in dimension 4. In order to do this we have carried out what is currently the largest scale simulations of the 4-dimensional Ising model, extending the lattices size up to $L=256$ and the number of samples per size by several orders of magnitude compared to earlier works, keeping track of data for both the canonical and microcanonical ensembles. Our conclusion is that the Ising model has a bounded specific heat, while the $ϕ^4$-model is known to have a logarithmic divergence at the critical point. Hence the two models belong to distinct universality classes in dimension 4.

cond-mat.stat-mech↗

arXiv2020

Efficient computation of permanents, with applications to boson sampling and random matrices

In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we first discuss how to compute the permanent efficiently on a parallel computer, followed by algorithms which provide an exponential speed-up for sparse matrices and linear run times for matrices of limited bandwidth. The parallel algorithm has been implemented in a freely available software package, also available in an efficient serial version. As part of the timing runs for this package we set a new world record for the matrix order on which a permanent has been computed. Next we perform a simulation study of several conjectures regarding the distribution of the permanent for random matrices. Here we focus on permanent anti-concentration conjecture, which has been used to find the classical computational complexity of Boson sampling. We find a good agreement with the basic versions of these conjectures and based on our data we propose refined versions of some of them. For small systems we also find noticable deviations from a propose strengthening of a bound for the number of photons in a Boson sampling system.

quant-ph↗

arXiv2020

Edge precoloring extension of hypercubes

We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most $d-1$ edges of the $d$-dimensional hypercube $Q_d$ can be extended to a proper $d$-edge coloring of $Q_d$. Additionally, we characterize which partial edge colorings of $Q_d$ with precisely $d$ precolored edges are extendable to proper $d$-edge colorings of $Q_d$, and consider some related edge precoloring extension problems of hypercubes.

math.CO↗

arXiv2018

Revisiting the cavity-method threshold for random 3-SAT

A detailed Monte Carlo-study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies $α_3 \leq 4.262$. If the assumption is correct, this means that the actual threshold value for $k=3$ is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large $k$. Our result thus indicate that there are distinct behaviors for $k$ above and below some critical $k_c$, and the cavity method may provide a correct mean-field picture for the range above $k_c$.

cond-mat.stat-mech↗

arXiv2017

Restricted extension of sparse partial edge colorings of hypercubes

We consider the following type of question: Given a partial proper $d$-edge coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors for the non-colored edges of $Q_d$,can we extend the partial coloring to a proper $d$-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.

math.CO↗

arXiv2016

The scaling window of the 5D Ising model with free boundary conditions

The five-dimensional Ising model with free boundary conditions has recently received a renewed interest in a debate concerning the finite-size scaling of the susceptibility near the critical temperature. We provide evidence in favour of the conventional scaling picture, where the susceptibility scales as $O(L^2)$ inside a critical scaling window of width $O(1/L^2)$. Our results are based on Monte Carlo data gathered on system sizes up to $L=79$ (ca. three billion spins) for a wide range of temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility and also the scaling and distribution of the size of the Fortuin-Kasteleyn cluster containing the origin. The probability of this cluster reaching the boundary determines the correlation length, and its behaviour agrees with the mean field critical exponent $δ=3$, that the scaling window has width $O(1/L^2)$.

cond-mat.stat-mech↗

arXiv2015

On 1-sum flows in undirected graphs

Let G=(V,E) be a simple undirected graph. For a given set L of the real line, a function omega from E to L is called an L-flow. Given a vector gamma whose coordinates are indexed by V, we say that omega is a gamma-L-flow if for each v in V, the sum of the values on the edges incident to v is gamma(v). If gamma(v)=c, for all v in V, then the gamma-L-flow is called a c-sum L-flow. In this paper we study the existence of gamma-L-flows for various choices of sets L of real numbers, with an emphasis on 1-sum flows. Given a natural k number, a c-sum k-flow is a c-sum flow with values from the set {-1,1,...,1-k, k-1}. Let L be a subset of real numbers containing 0 and let L* be L minus 0 by L*. Answering a question from a recent paper we characterize which bipartite graphs admit a 1-sum R*-flow or a 1-sum Z*-flow. We also show that that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.

math.CO↗

arXiv2015

The discontinuity of the specific heat for the 5D Ising model

In this paper we investigate the behaviour of the specific heat around the critical point of the Ising model in dimension 5 to 7. We find a specific heat discontinuity, like that for the mean field Ising model, and provide estimates for the left and right hand limits of the specific heat at the critical point. We also estimate the singular exponents, describing how the specific heat approaches those limits. Additionally, we make a smaller scale investigation of the same properties in dimension 6 and 7, and provide strongly improved estimates for the critical termperature $K_c$ in $d=5,6,7$ which bring the best MC-estimate closer to those obtained by long high temperature series expanions.

cond-mat.stat-mech↗

arXiv2014

Finite size scaling of the 5D Ising model with free boundary conditions

There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives a $L^2$ scaling for the susceptibility and an alternative theory has promoted a $L^{5/2}$ scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to side $L=160$ and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provides both supports the standard scaling picture and provides a clear explanation of why the scalings for free and cyclic boundary should be different.

cond-mat.stat-mech↗

arXiv2014

Complete graph asymptotics for the Ising and random cluster models on 5D grids with cyclic boundary

The finite size scaling behaviour for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject for a long running debate. The older papers have been based on ideas from e.g. field theory or renormalization. In this paper we propose a detailed and exact scaling picture for critical region of the model with cyclic boundary. Unlike the previous papers our approach is based on a comparison with the existing exact and rigorous results for the FK-random-cluster model on a complete graph. Based on those results we identify several distinct scaling regions in an $L$-dependent window around the critical point. We test these predictions by comparing with data from Monte Carlo simulations and find a good agreement. The main feature which differs between the complete graph and the five dimensional model with free boundary is the existence of a bimodal energy distribution near the critical point in the latter. This feature was found by the same authors in an earlier paper in the form of a quasi-first order phase transition for the same Ising model.

cond-mat.stat-mech↗

arXiv2012

Critical behaviour of the Ising model on the 4-dimensional lattice

In this paper we investigate the nature of the singularity of the Ising model of the 4-dimensional cubic lattice. It is rigorously known that the specific heat has critical exponent $α=0$ but a non-rigorous field-theory argument predicts an unbounded specific heat with a logarithmic singularity at $T_c$. We find that within the given accuracy the canonical ensemble data is consistent both with a logarithmic singularity and a bounded specific heat, but that the micro-canonical ensemble lends stronger support to a bounded specific heat. Our conclusion is that either much larger system sizes are needed for Monte Carlo studies of this model in four dimensions or the field theory prediction of a logarithmic singularity is wrong.

cond-mat.stat-mech↗

arXiv2010

Non-vanishing boundary effects and quasi-first order phase transitions in high dimensional Ising models

In order to gain a better understanding of the Ising model in higher dimensions we have made a comparative study of how the boundary, open versus cyclic, of a d-dimensional simple lattice, for d=1,...,5, affects the behaviour of the specific heat C and its microcanonical relative, the entropy derivative -dS/dU. In dimensions 4 and 5 the boundary has a strong effect on the critical region of the model and for cyclic boundaries in dimension 5 we find that the model displays a quasi first order phase transition with a bimodal energy distribution. The latent heat decreases with increasing systems size but for all system sizes used in earlier papers the effect is clearly visible once a wide enough range of values for K is considered. Relations to recent rigorous results for high dimensional percolation and previous debates on simulation of Ising models and gauge fields are discussed.

cond-mat.stat-mech↗

arXiv2008

On the Number of Matchings in Regular Graphs

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for 2-regular bipartite graphs and for $m$-matchings with $m\le 4$.

math.CO↗

arXiv2006

Reconstruction of the finite size canonical ensemble from incomplete micro-canonical data

In this paper we discuss how partial knowledge of the density of states for a model can be used to give good approximations of the energy distributions in a given temperature range. From these distributions one can then obtain the statistical moments corresponding to eg the internal energy and the specific heat. These questions have gained interest apropos of several recent methods for estimating the density of states of spin models. As a worked example we finally apply these methods to the 3-state Potts model for cubic lattices of linear order up to 128. We give estimates of eg latent heat and critical temperature, as well as the microcanonical properties of interest.

cond-mat.stat-mech↗