107 research outputs found
Locality of connective constants
The connective constant of a quasi-transitive graph is the
exponential growth rate of the number of self-avoiding walks from a given
origin. We prove a locality theorem for connective constants, namely, that the
connective constants of two graphs are close in value whenever the graphs agree
on a large ball around the origin (and a further condition is satisfied). The
proof exploits a generalized bridge decomposition of self-avoiding walks, which
is valid subject to the assumption that the underlying graph is
quasi-transitive and possesses a so-called unimodular graph height function
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
Critical Surface of the Hexagonal Polygon Model
The hexagonal polygon model arises in a natural way via a transformation of the 1–2 model on the hexagonal lattice, and it is related to the high temperature expansion of the Ising model. There are three types of edge, and three corresponding parameters α,β,γ>0. By studying the long-range order of a certain two-edge correlation function, it is shown that the parameter space (0,∞)3 may be divided into subcritical and supercritical regions, separated by critical surfaces satisfying an explicitly known formula. This result complements earlier work on the Ising model and the 1–2 model. The proof uses the Pfaffian representation of Fisher, Kasteleyn, and Temperley for the counts of dimers on planar graphs.This work was supported in part by the Engineering and Physical Sciences Research Council under Grant EP/I03372X/1. ZL acknowledges support from the Simons Foundation under Grant #351813
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Self-avoiding walks and amenability
The connective constant of an infinite transitive graph is the
exponential growth rate of the number of self-avoiding walks from a given
origin. The relationship between connective constants and amenability is
explored in the current work.
Various properties of connective constants depend on the existence of
so-called 'graph height functions', namely: (i) whether is a local
function on certain graphs derived from , (ii) the equality of and
the asymptotic growth rate of bridges, and (iii) whether there exists a
terminating algorithm for approximating to a given degree of accuracy.
In the context of amenable groups, it is proved that the Cayley graphs of
infinite, finitely generated, elementary amenable groups support graph height
functions, which are in addition harmonic. In contrast, the Cayley graph of the
Grigorchuk group, which is amenable but not elementary amenable, does not have
a graph height function.
In the context of non-amenable, transitive graphs, a lower bound is presented
for the connective constant in terms of the spectral bottom of the graph. This
is a strengthening of an earlier result of the same authors. Secondly, using a
percolation inequality of Benjamini, Nachmias, and Peres, it is explained that
the connective constant of a non-amenable, transitive graph with large girth is
close to that of a regular tree. Examples are given of non-amenable groups
without graph height functions, of which one is the Higman group
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Critical surface of the 1-2 model
The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. There are three edge directions, and three corresponding parameters a, b, c. It is proved that, when a ≥ b ≥ c >0 , the surface given by √a=√b+√c is critical. The proof hinges upon a representation of the partition function in terms of that of a certain dimer model. This dimer model may be studied via the Pfaffian representation of Fisher, Kasteleyn, and Temperley. It is proved, in addition, that the two-edge correlation function converges exponentially fast with distance when √a≠√b+√c. Many of the results may be extended to periodic models.This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1. Z.L.’s research was supported by the Simons Foundation grant # 351813 and National Science Foundation DMS-1608896. We thank the referee for a detailed and useful report
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Connective constants and height functions for Cayley graphs
The connective constant () of an infinite transitive graph is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called “unimodular graph height functions”. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a “ height function”. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency.
It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic.
Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.The first author was supported in part by EPSRC Grant EP/I03372X/1. The second author was supported in part by Simons Collaboration Grant #351813 and NSF grant #1608896
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications
[EN] This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR-type epidemiological models based on discrete-time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.Spanish Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-P; Generalitat Valenciana, Grant/Award Number: APOSTD/2019/128; Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-PCortés, J.; El-Labany, S.; Navarro-Quiles, A.; Selim, MM.; Slama, H. (2020). A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications. Mathematical Methods in the Applied Sciences. 43(14):8204-8222. https://doi.org/10.1002/mma.6482S820482224314Hamra, G., MacLehose, R., & Richardson, D. (2013). Markov Chain Monte Carlo: an introduction for epidemiologists. International Journal of Epidemiology, 42(2), 627-634. doi:10.1093/ije/dyt043Becker, N. (1981). A General Chain Binomial Model for Infectious Diseases. Biometrics, 37(2), 251. doi:10.2307/2530415Allen, L. J. S. (2010). An Introduction to Stochastic Processes with Applications to Biology. doi:10.1201/b12537Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599-653. doi:10.1137/s0036144500371907Brauer, F., & Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics. doi:10.1007/978-1-4757-3516-1Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Some results about randomized binary Markov chains: theory, computing and applications. International Journal of Computer Mathematics, 97(1-2), 141-156. doi:10.1080/00207160.2018.1440290Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics, 324, 225-240. doi:10.1016/j.cam.2017.04.040Casabán, M.-C., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 86-97. doi:10.1016/j.cnsns.2014.12.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Slama, H., Hussein, A., El-Bedwhey, N. A., & Selim, M. M. (2019). An approximate probabilistic solution of a random SIR-type epidemiological model using RVT technique. Applied Mathematics and Computation, 361, 144-156. doi:10.1016/j.amc.2019.05.019Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Van den Driessche, P. (2017). Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2(3), 288-303. doi:10.1016/j.idm.2017.06.002Heffernan, J. ., Smith, R. ., & Wahl, L. . (2005). Perspectives on the basic reproductive ratio. Journal of The Royal Society Interface, 2(4), 281-293. doi:10.1098/rsif.2005.0042Khalil, K. M., Abdel-Aziz, M., Nazmy, T. T., & Salem, A.-B. M. (2012). An Agent-Based Modeling for Pandemic Influenza in Egypt. Intelligent Systems Reference Library, 205-218. doi:10.1007/978-3-642-25755-1_1
Metastability in the dilute Ising model
Consider Glauber dynamics for the Ising model on the hypercubic lattice with
a positive magnetic field. Starting from the minus configuration, the system
initially settles into a metastable state with negative magnetization. Slowly
the system relaxes to a stable state with positive magnetization. Schonmann and
Shlosman showed that in the two dimensional case the relaxation time is a
simple function of the energy required to create a critical Wulff droplet.
The dilute Ising model is obtained from the regular Ising model by deleting a
fraction of the edges of the underlying graph. In this paper we show that even
an arbitrarily small dilution can dramatically reduce the relaxation time. This
is because of a catalyst effect---rare regions of high dilution speed up the
transition from minus phase to plus phase.Comment: 49 page
First-Hitting Times Under Additive Drift
For the last ten years, almost every theoretical result concerning the
expected run time of a randomized search heuristic used drift theory, making it
the arguably most important tool in this domain. Its success is due to its ease
of use and its powerful result: drift theory allows the user to derive bounds
on the expected first-hitting time of a random process by bounding expected
local changes of the process -- the drift. This is usually far easier than
bounding the expected first-hitting time directly.
Due to the widespread use of drift theory, it is of utmost importance to have
the best drift theorems possible. We improve the fundamental additive,
multiplicative, and variable drift theorems by stating them in a form as
general as possible and providing examples of why the restrictions we keep are
still necessary. Our additive drift theorem for upper bounds only requires the
process to be nonnegative, that is, we remove unnecessary restrictions like a
finite, discrete, or bounded search space. As corollaries, the same is true for
our upper bounds in the case of variable and multiplicative drift
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