اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPhase transitions and percolation at criticality in enhanced random connection models
120
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $mathcal{P}_{lambda}$ in $mathbb{R}^2$ of intensity $lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(|x-y|)$, independent of everything else, where $g:[0,infty) to [0,1]$ and $| cdot |$ is the Euclidean norm. In the inhomogenous version of the model, points of $mathcal{P}_{lambda}$ are endowed with weights that are non-negative independent random variables with distribution $P(W>w)= w^{-beta}1_{[1,infty)}(w)$, $beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability $1 - expleft( - frac{eta W_xW_y}{|x-y|^{alpha}} right)$, $eta, alpha > 0$, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of $mathcal{P}_{lambda}$. A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of $mathcal{P}_{lambda}$. Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.
قيم البحث
اقرأ أيضاً
We investigate the geometric properties of loops on two-dimensional lattice graphs, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of spanning loops of total negativ
e weight. The resulting percolation problem is fundamentally different from conventional percolation, as we have seen in a previous study of this model for the undiluted case. Here, we investigate how the percolation transition is affected by additional dilution. We consider two types of dilution: either a certain fraction of edges exhibit zero weight, or a fraction of edges is even absent. We study these systems numerically using exact combinatorial optimization techniques based on suitable transformations of the graphs and applying matching algorithms. We perform a finite-size scaling analysis to obtain the phase diagram and determine the critical properties of the phase boundary. We find that the first type of dilution does not change the universality class compared to the undiluted case whereas the second type of dilution leads to a change of the universality class.
We study a class of models of i.i.d.~random environments in general dimensions $dge 2$, where each site is equipped randomly with an environment, and a parameter $p$ governs the frequency of certain environments that can act as a barrier. We show tha
t many of these models (including some which are non-monotone in $p$) exhibit a sharp phase transition for the geometry of connected clusters as $p$ varies.
Using the randomized algorithm method developed by Duminil-Copin, Raoufi, Tassion (2019b) we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof that the critical parameter for percolation in this model
is $1/2$ when the underlying shapes for the distinct colours arise from the same distribution and extends the work of Hirsch (2015) and M{u}ller (2016). In addition we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain `transitivity condition this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.
We investigate a large class of random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight
and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions $din{1,2}$ we completely characterise recurrence vs transience of random walks on the infinite cluster. In $dgeq 3$ we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. Our results are particularly interesting for the special case of the age-dependent random connection model recently introduced in [P. Gracar et al., The age-dependent random connection model, Queueing Syst. {bf 93} (2019), no.~3-4, 309--331. MR4032928].
In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two dimensional lattices, whose transitions involve non-local or topological properties, including site and bond percolations,
the XY model and the generalized XY model. We find that using just one hidden layer in a fully-connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent $ u$. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects---vortices and anti-vortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter $Delta$ values, where $Delta$ is the relative weight of pure XY coupling. Besides using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size $Ltimes L$ can be used to the phase diagram for other sizes ($Ltimes L$, where $L e L$) without any further training.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
