Haut PDF Nesting statistics in the $O(n)$ loop model on random planar maps

Nesting statistics in the $O(n)$ loop model on random planar maps

Nesting statistics in the $O(n)$ loop model on random planar maps

2.4. Main results on random maps. This paper is concerned with the statistical prop- erties of nesting between loops. The situation is simpler in the planar case since every loop is contractible, and divides the underlying surface into two components. The nesting structure of large maps of arbitrary topology will be analyzed in a forthcoming work [15]. In the general O(n) loop model, the generating series of disks and cylinders have been characterized in [12, 11, 14], and explicitly computed in the model with bending energy in [11], building on the previous works [40, 13]. This characterization is a linear functional relation which depends explicitly on n, accompanied by a non-linear consistency relation depending implicitly on n. We remind the steps leading to this characterization in Sec- tions 3-4. In particular, we review in Section 3 the nested loop approach developed in [12], which allows enumerating maps with loop configurations in terms of generating series of usual maps. We then derive in Section 4 the functional relations for maps with loops as direct consequences of the well-known functional relations for generating series of usual maps. The key to our results is the derivation in Section 4.4 of an analog characterization for refined generating series of pointed disks (resp. cylinders), in which the loops which separates the origin (resp. the second boundary) and the (first) boundary face are counted with an extra weight s each. We find that the characterization of the generating series is only modified by replacing n with ns in the linear functional relation, while keeping n in the consistency relation. Subsequently, in the model with bending energy, we can compute explicitly the refined generating series, in Section 5. We analyze in Section 6 the behavior of those generating series at a non-generic critical point which pertains to the O(n) model. In the process, we rederive the phase diagram of the model with bending energy, and we eventually find:
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Central loops in random planar graphs

Central loops in random planar graphs

Benjamin Lion and Marc Barthelemy ∗ Institut de Physique Th´ eorique, CEA, CNRS-URA 2306, F-91191, Gif-sur-Yvette, France Random planar graphs appear in a variety of context and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display non-trivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of non-trivial patterns such as central loops. This toy model is made of a star network with N b branches of size n and links of weight 1, superimposed to a loop at distance ` from
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Random Planar Maps coupled to Spin Systems

Random Planar Maps coupled to Spin Systems

by statistical physics models. We examine three particular models using tools coming from analysis, combinatorics and probability. From a geometric perspective, we focus on the interface properties and the local limits of the decorated random maps. The first model defines a family of random quadrangulations of the disk decorated by an O(n)-loop model. After completing the proof of its phase diagram initiated in [ BBG12c ] (Chap. II ), we look into the lengths and the nesting structure of the loops in the non-generic critical phase (Chap. III ). We show that these statistics, described as a labeled tree, converge in distribution to an explicit multiplicative cascade when the perimeter of the disk tends to infinity. The second model (Chap. IV ) consists of random planar maps decorated by the Fortuin-Kasteleyn percolation. We complete the proof of its local convergence sketched in [ She16b ] and establish a number of properties of the limit. The third model (Chap. V ) is that of random triangulations of the disk decorated by the Ising model. It is closely related to the O(n)-decorated quadrangulation when n = 1. We compute explicitly the partition function of the model with Dobrushin boundary conditions at its critical point, in a form ameneable to asymptotics. Using these asymptotics, we study the peeling process along the Ising interface in the limit where the perimeter of the disk tends to infinity.
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Random matrices and the Potts model on random graphs

Random matrices and the Potts model on random graphs

In these lecture notes we discuss the uses of topological expansions to study the Potts model on random graphs, that is some (shaded) elaboration of the so-called O(n) model. We will start by describing the topological expansion and how they can be derived from Gaussian calculus, that is Wick formula. To enumerate interesting graphs, that is graphs with several vertices, it turns out that one needs to compute Laplace transforms of traces of polynomials of random matrices. Such integrals might then diverge and their relations with maps enumeration be only formal. We shall discuss the relation between topological expansions and matrix integrals via a third mathematical concept, that is the so-called loop (or Schwinger-Dyson’s) equations. These equations are simply derived from the matrix integrals via integration by parts. They correspond to induction relation in the enumeration of maps, as first introduced by Tutte to enumerate planar graphs. We shall describe how these equations give an alternative bridge between matrix integrals and the enumeration of maps which in fact allows to turn formal equalities into asymptotic equalities. Based on this point, one can define properly matrix integrals whose asymptotics are given by generating functions for the enumeration of planar connected graphs. Finally, we will specialize this relation to analyze the Potts model on certain random planar graphs. This connection is based on the interpretation of the Potts model as a loop model. Loop models naturally appear in a variety of statistical models where the loops represent the configuration of boundaries of some random regions. Perhaps the most famous of these is the so-called O(n) loop model which can be described as follows.
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On the growth of random planar maps with a prescribed degree sequence

On the growth of random planar maps with a prescribed degree sequence

1.1. Model and notation. — Recall that a (rooted planar) map is a finite (multi-)graph embedded in the two-dimensional sphere, in which one oriented edge is distinguished (the root-edge), and viewed up to orientation-preserving homeomorphisms to make it a discrete object. The embedding allows to define the faces of the map which are the connected components of the complement of the graph on the sphere, and the degree of a face is the number of edges incident to it, counted with multiplicity: an edge incident on both sides to the same face contributes twice to its degree. For technical reasons, we restrict ourselves to bipartite maps, in which all faces have even degree. The face incident to the right of the root-edge is called the root-face, whereas the other faces are called inner faces. The collection of edges incident to the root-face is the boundary of the map. A map with only two boundary edges can be seen as a map without boundary by gluing these two edges together.
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Bipolar orientations on planar maps and SLE12

Bipolar orientations on planar maps and SLE12

duality.) These special {κ, κ 0 } pairs include {2, 8} (for loop-erased random walk and the uniform spanning tree) [ LSW04 ], {8/3, 6} (for percolation and Brownian motion) [ Smi01 , LSW01a ], {3, 16/3} (for the Ising and FK-Ising model) [ Smi10 , CDCH + 14 ], and {4, 4} (for the Gaussian free field contours) [ SS09 , SS13 ]. The relationships between these special {κ, κ 0 } values and the corresponding discrete models were all discovered or conjectured within a couple of years of Schramm’s introduction of SLE, building on earlier arguments from the physics literature. We note that all of these relationships have random planar map analogs, and that they all correspond to {κ, κ 0 } ⊂ [2, 8]. This range is significant because the so-called conformal loop ensembles CLE κ [ She09 , SW12 ] are only defined for κ ∈ (8/3, 8], and the discrete models mentioned above are all related to random collections of loops in some way, and hence have either κ or κ 0 in the range (8/3, 8]. Furthermore, it has long been
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Scaling limits of random trees and planar maps

Scaling limits of random trees and planar maps

connections between planar maps and the statistical physics of random surfaces can be found in Bouttier’s thesis [7]. From the probabilistic perspective, a planar map can be viewed as a discretization of a surface, and finding a continuous limit for large planar maps chosen at random in a suitable class should lead to an interesting model of a “Brownian surface”. This is of course analogous to the well-known fact that Brownian motion appears as the scaling limit of long discrete random paths. In a way similar to the convergence of rescaled random walks to Brownian motion, one expects that the scaling limit of large random planar maps is universal in the sense that it should not depend on the details of the discrete model one is considering. These ideas appeared in the pioneering paper of Chassaing and Schaeffer [12] and in the subsequent work of Markert and Mokkadem [37] in the case of quadrangulations, and a little later in Schramm [48], who gave a precise form to the question of the existence of a scaling limit for large random triangulations of the sphere.
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Refined universal laws for hull volumes and perimeters in large planar maps

Refined universal laws for hull volumes and perimeters in large planar maps

1. Introduction The study of random planar maps, which are connected graphs embedded on the sphere, has been for more than fifty years the subject of some intense activity among combinatori- alists and probabilists, as well as among physicists in various domains. Very recently, some special attention was paid to statistical properties of the hull in random planar maps, a problem which may be stated as follows: consider an ensemble of planar maps having two marked vertices at graph distance k from each other. For any non-negative d strictly less than k, we may find a closed line “at distance d” (i.e. made of edges connecting vertices at distance d or so) from the first vertex and separating the two marked vertices from each other. Several prescriptions may be adopted for a univocal definition of this separating line but they all eventually give rise to similar statistical properties. The separating line divides de facto the map into two connected components, each containing one of the marked ver- tices. The hull at distance d corresponds to the part of the map lying on the same side as the first vertex (i.e. that from which distances are measured). The geometrical characteristics of this hull for arbitrary d and k provide random variables whose statistics may be studied by various techniques. In particular, the statistics of the volume of the hull. which is its number of faces, and of the hull perimeter, which is the length of the separating line, have been the subject of several investigations [11, 10, 5, 4, 9, 12, 8].
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Infinite random planar maps related to Cauchy processes

Infinite random planar maps related to Cauchy processes

1 Introduction This work studies the geometry of random Boltzmann planar maps with high degrees in the continuation of [ 10 ] and focuses on the critical case a = 2 which was left aside there. The geometry of those maps turns out to involve in an intricate way random walks whose step distribution is in the domain of attraction of the standard symmetric Cauchy process and displays an unexpected large scale geometry such as an intermediate rate of growth. Let us first present rigorously the model of random maps we are dealing with. To stick to the existing literature we prefer to introduce the model of maps with large faces and then take their dual maps to get maps with large vertex degrees (as opposed to dealing with maps with large vertex degrees directly).
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The two-point function of bicolored planar maps

The two-point function of bicolored planar maps

1. Introduction Distance properties within planar maps have raised a lot of interest in the recent years and led to many remarkable results on the statistics of distance correlations within families of random maps. Still many questions are not yet solved, and many improvements of the present results, although quite natural, remain challenging. One of the simplest characterization of the distance statistics within maps is probably the distance-dependent two-point function which, roughly speaking, enumerates maps with two “points” (typically edges or vertices) at a fixed given graph distance within the map. Such two-point functions were first computed in [3] for general families of bipartite planar maps with controlled face degrees (including the simplest case of quadrangulations). Although this is not quite the method used in [3], it has now become clear that the simplest way to get two-point functions is via a distance- preserving bijection between maps and tree-like objects called mobiles, originally found by Schaeffer [17, 10] (rephrasing a bijection due by Cori and Vauquelin [11]) in the case of quadrangulations and later generalized to the case of arbitrary maps [5]. More recently, a similar bijection extending Schaeffer’s ideas, due to Ambjørn and Budd, has made it possible to compute the two-point function of maps with arbitrary large face degrees, controlled by both their number of edges and faces [2], and more generally that of bipartite maps or hypermaps [6] with arbitrarily large face degrees.
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Some results on the statistics of hull perimeters in large planar triangulations $and$ quadrangulations

Some results on the statistics of hull perimeters in large planar triangulations $and$ quadrangulations

1. Introduction Understanding the statistics of random planar maps, i.e. connected graphs embedded on the sphere, as well as their various continuous limits, such as the Brownian map [9, 8] or the Brownian plane [1], is a very active field of both combinatorics and probability theory. In this paper, we study the statistics of hull perimeters in large planar maps, a problem which may heuristically be understood as follows: consider a planar map M of some type (in the following, we shall restrict our analysis to the case of triangulations and quadrangulations) with two marked vertices, an origin vertex v 0 and a second distinguished vertex v 1 at graph distance k from v 0 , for some k ≥ 2. Consider now, for some d strictly between 0 and k, the ball of radius d which, so to say, is the part of the map at graph distance less than d from the origin 1 . This ball has a boundary made in general of several closed lines, each line linking vertices at distance of order d from v 0 and separating v 0 from a connected domain where all vertices are at distance larger than d (see figure 1). One of these domains C d contains the second distinguished vertex v 1 and we may define the hull of radius d as the domain H d = M\C d , namely the union of the ball of radius d itself and of all the connected domains at distance larger than d which do not contain v 1 (see figure 1 where H d is represented in light blue). The hull boundary is then the boundary of H d (which is also that of C d ), forming a closed line at distance d from v 0 and separating v 0 from v 1 . The length of this boundary is called the hull perimeter at distance d and will be denoted by L(d) in the following.
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Monte Carlo particle transport in random media: The effects of mixing statistics

Monte Carlo particle transport in random media: The effects of mixing statistics

185 ables, in the limit case of domains having an infinite exten- sion (4; 5; 35). In this respect, Poisson tessellations have been shown to pos- sess a remarkable property: in the limit of infinite domains, an arbitrary line will be cut by the hyperplanes of the tessel- lation into chords whose lengths are exponentially distributed with parameter ρ P (whence the identification with Markovian mixing). Thus, in this case, the average chord length h Λi ∞ sat- isfies h Λi ∞ = ρ −1 P , and its probability density Π
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Statistics of geometric random simplicial complexes

Statistics of geometric random simplicial complexes

illustrated in Fig. 1, representing the case where a point is deployed over a plan. One of the main results of this paper is the explicit expression for the variance of the number of k-simplices, N k , the covariance between N k and N l and the variance of the Eulers’s characteristic, χ, in such complex, which allows us to apply concen- trations inequalities. For d ≥ 2, χ is expressed by a power serie and if d = 1, it is possible to find its closed-form expression. A complex closed-form expression for the third order central moment of N k is explicitely calculated and using the same
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Statistics of localized phase slips in tunable width planar point contacts

Statistics of localized phase slips in tunable width planar point contacts

Thus far, most of the investigations have focussed on long nanowires 15,16 where the loci of the events along the nanowire are undetermined, or on short suspended nanowires with poor heat removal conditions 6,17,18 where single PS events may have more dramatic consequences. In the present report, we explore the opposite situation of a point contact between two Al superconducting banks forcing the appearance of PS to be highly localized in space. In addition, with the Al film being deposited on a Si/SiO 2 single crystal wafer, heat removal is rather effi- cient. The possibility to narrow down the size of the constriction in-situ through controlled migration of atoms, allows us to switch, on the very same sample, between strong and weak self-heating regimes after the switching process. Furthermore, an analysis of the statistical distribution of I sw for different temperatures reveals that the main dissipative mechanism can be ascribed to a train of thermally driven PS events. The physics addressed in the present manuscript might provide a new approach for understanding the statistics of flux avalanche triggering of thermomagnetic origin in thin superconducting films or the escape distribution for magnetization reversal in small particles, although sampling events distribution might be harder in these systems.
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Exploring the Cloud Computing Loop in the Strategic Alignment Model

Exploring the Cloud Computing Loop in the Strategic Alignment Model

At the second stage, Utilization of Information features the outcomes of cloud adop- tion and performance. At this stage, IT and business strategy can be designed reflecting on information received at stage one through optimizing and implementing strategy. It demands decision makers to apply knowledge obtained at the first stage to design strat- egy. They may either design a brand-new strategy, or update the existing strategy in light of changes in business processes and IT infrastructure triggered by CC. At this stage optimisation and implementation are required. Optimisation gains understanding of CC for organizational and IT strategy. This results in developing practical solutions and plans from abstract ideas, trends and insights. Given a well-defined solution, deci- sion makers should be able to sort through large amounts of information to pinpoint the critical factors and processes where cloud is required. They should be confident in their ability to make a sound, logical evaluation of transformative impact of CC and integrate those processes. Finally, implementation is execution of designed strategy. Implemen- tation of strategy requires complete understanding of how business operations and IT infrastructure are going to be affected by the CC. In case of strategy complete or partial failure or as a result of changes in external environment (e.g. technology, institutions, market competition, etc.), decision makers need to be agile and respond quickly. This requires transition from the second to the third stage of “Validated Learning”.
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Statistics of geometric random simplicial complexes

Statistics of geometric random simplicial complexes

not possible to find as much results as found for the one-dimensional case, but the results in this work holds for any dimension d. The principal idea of the problem is that each sensor can control some environ- mental information (such as temperature, pression, presence of an intruder, etc) around them. The homology of the coverage of this network, as shown in [5], can be represented by a simplicial complex. A simplicial complex is a generalization of a graph, so, while we represent a graph with points and edges, a simplicial complex can be represented by points, edges, filled triangles, filled tetraedrons and so on. Almost all the work considers the radius of monitoring ǫ but a different interpreta- tion can be done if the sensors are communicating amoung them. In this case, we suppose that sensors have a power suply allowing them to transmit theirs ID’s and, at the same time, sensors have receivers which can identify the transmitted ID’s of other sensors above a threshold power. The sensors, knowing mutually the ID’s of the close neighbors, are considered connected, creating an information network. The problem remains analogous as the previous one, except that we substitute the coverage radius ǫ by a communication one of ǫ/2. We can see examples of sim- plicial complexes representations given by sensors communicating amoung them or monitoring a region in Fig. 1.
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Planar maps, circle patterns and 2d gravity

Planar maps, circle patterns and 2d gravity

1 Introduction It has been argued by physicists [David, 1985] [Fröhlich, 1985] [Kazakov, 1985] , that the continuous limit of large 2-dimensional maps, should be the same thing as the so-called 2d-quantum gravity, i.e. a theory of random surfaces, or also a theory of random metrics (for general references on the subject see e.g. [Ambjørn et al., 2005]). For 2d quantum gravity, since changing the metrics can be partially absorbed by reparametrizing the surface, Polyakov first proposed to gauge out the diffeomorphism group, by chosing a conformal metrics [Polyakov, 1981]. The measure (probability weight) of the conformal metrics, is then the Jacobian of the gauge fixing operator, which itself can be written as a Gaussian integral over Fadeev-Popov ghosts, with the quadratic form encoding
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Combinatorics of planar maps and algorithmic applications.

Combinatorics of planar maps and algorithmic applications.

5. Schnyder woods W. Schnyder has introduced in [101] a beautiful combinatorial structure for triangulations, which he called realizer, and is now usually called Schnyder wood. Several equivalent definitions have been given in terms of angle labelings, orien- tations, or colorings. The concept has also a natural extension to 3-connected maps [3, 46]. Essentially a Schnyder wood is an orientation and partition of the inner edges of a triangulation into three trees that are directed to each of the three outer vertices. The definition for 3-connected maps is similar, expect that an edge is allowed to be traversed by either one tree or by two trees traversing the edge in opposite directions. Originally, Schnyder woods have been introduced to obtain a nice characterization of planarity in terms of a “dimension” parameter associated with the incidences vertices-edges [101]. These properties have beautiful geometric interpretations in terms of orthogonal surfaces in the 3D space, see [47, 88, 119]. Schnyder woods also lead to an elegant straight-line drawing algorithm based on barycentric representation of vertices, also due to Schnyder [102] and subsequently improved by Zhang and He [118] and by Bonichon et al [16] (see Chapter 5 for a presentation and analysis of the latter algorithm). In this section we define Schny- der woods and state their main properties, in particular the decomposition into three spanning trees. We also describe the lattice structure, following the study by Brehm [25] for triangulations and Felsner [48] for 3-connected maps.
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Periodicity of certain piecewise affine planar maps

Periodicity of certain piecewise affine planar maps

show that no point with aperiodic trajectory has coordinates in Z[λ], which proves Conjecture 1.1 for these eight values of λ. This number theoretical problem is solved by introducing a map S, which is the composition of the first hitting map to the image of a suitably chosen self inducing domain under a (contracting) scaling map and the inverse of the scaling map. A crucial fact is that the inverse of the scaling constant is a Pisot unit in the quadratic number field Q(λ). This number theoretical argument greatly reduces the classification problem of periodic orbits, see e.g. Theorem 2.1. All possible period lengths can be determined explicitly and one can even construct concrete aperiodic points
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Spectral statistics for weakly correlated random potentials

Spectral statistics for weakly correlated random potentials

Note that now the “local” Hamiltonians (H ω,Λ ˜ ℓ (γ j ) ) j are not necessarily stochastically independent. In our application of Theorem 2.4 to the model (1.1), the choice we will make for the parameters α, β ′ , β and ˜ ρ will be quite different from the one made in [6]. While in [6] the authors wanted to maximize the admissible size f |I Λ (i.e. minimize α), here, our primary concern will be to control the

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