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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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**