To help the reader, we now give a quick summary of our results and notations. The **boundary** **loop** model (BLM) to be studied is defined on a tilted square lattice (see Fig. 3), wrapped on an annulus of width N strands and circumference M lattice spacings. Loops cover all the edges, and interact in a specific way with the outer rim of the annulus, whereas they are simply reflected by the inner rim (free **boundary** conditions). We denote by L the number of non contractible loops (note that L and N must have the same parity). Any **loop** has one of four weights (x, y, l or m, see Fig. 8): l (resp. m) for a non contractible **loop** never touching (resp. touching at least once) the outer rim, and similarly x (resp. y) for contractible loops. We parametrize x = q + q −1 ∈ (−2, 2] by q = e iπ/(p+1) (p real); the model is then critical with central charge (2.7) for any real values of y, l, m and is endowed with the Uq(sl2) quantum group symmetry. We further parametrize y = y(r) as in (2.4). Our central claim is that for any real r, and any L, there are two (distinct for L > 0) **conformal** **boundary** conditions: blobbed (resp. unblobbed) in which the outermost non contractible **loop** is required to (resp. required not to) touch the outer rim of the annulus. (When L = 0 the two cases coincide.) The spectrum generating functions in these two cases are (3.8), and the **boundary** **conformal** weights (critical exponents) hr,r±L are read off from (2.8). They combine to form the BLM partition function Z through the amplitudes (3.11). When p ≥ 1 is integer, and when further r = 1, 2, . . . , p the BLM model can be related to an RSOS model of the Ap type with specific **boundary** conditions (three columns of fixed heights, see Fig. 12) through the rules (4.4). In the latter case, Z can be written as a sum (3.18) over irreducible representations of the Virasoro algebra.

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of deep links with the powerful SLE approach [8]. It is therefore no surprise that the issue of CBC for **loop** **models** should be a major problem. This issue has however been slow to evolve, in part for technical reasons: the Coulomb gas formalism, which is so successful in the bulk case, is very difficult to carry out in the presence of boundaries, for not entirely clear reasons [9, 4]. It took progress on the algebraic side—through the study of **boundary** algebras and spin **models** with general **boundary** fields—for the simplest families of CBC to even be identified properly. The works [10, 11] finally showed that CBC were obtained in the dense **loop** model by simply giving to loops touching the **boundary** a fugacity n 1 different from the one in the bulk. Associated **conformal**

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Los Angeles, CA 90089, USA January 7, 2008
Abstract
We discuss in this paper combinatorial aspects of **boundary** **loop** **models**, that is **models** of self- avoiding loops on a strip where loops get different weights depending on whether they touch the left, the right, both or no **boundary**. These **models** are described algebraically by a generalization of the Temperley-Lieb algebra, dubbed the two-**boundary** TL algebra. We give results for the dimensions of TL representations and the corresponding degeneracies in the partition functions. We interpret these results in terms of fusion and in the light of the recently uncovered A n large symmetry present in **loop** **models**, paving the way for the analysis of the **conformal** field theory properties. Finally, we propose conjectures for determinants of Gram matrices in all cases, including the two-**boundary** one, which has recently been discussed by de Gier and Nichols.

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Although there is no general result relating integrability and criticality of a model, they can sometimes coincide. Integrable points often play some particular role in the phase diagram of a model, and in general the information they provide is a key point to understand the critical behaviour. In the particular case of **loop** **models**, some signs of a deeper relation between integra- bility and criticality have appeared very recently in the literature [26, 27]. The relation between integrability in the sense of Yang-Baxter and the lattice holomorphicity of certain discrete ob- servables, which would lead to critical **models** in the continuum limit, is a fascinating subject. We hope that future work will develop this approach and provide a straightforward and rigorous way to relate lattice parameters to the characteristics of the objects showing up in the continuum limit [28]. But for now, let us use the more traditional way of studying a lattice model: we look for (bulk and **boundary**) integrable solutions, and conjecture that they correspond to critical points of the **loop** model.

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The observations on the braid translation in the gl(1|1) case are generalized to the case of the sl (2|1) spin chain (which is related with the statistics of percolation hulls, and a LCFT with central charge c = 0 [20, 26, 42, 52]) with totally similar results.
These results, in hindsight, are somewhat natural. Braid translation works for minimal mod- els because there are so few irreducible representations of the blob and periodic Temperley-Lieb algebras. In contrast, in the logarithmic case, non semi-simplicity guarantees a great variety of possible representations. The experience gathered in the last few years [17, 19, 20] suggests that the representations contributing to periodic **models**, in the logarithmic case, are considerably more complicated than those known so far to occur in the open case [48]. It is thus not surprising that braid translation, in our present state of knowledge, does not allow us to connect straightforwardly **models** with open and periodic **boundary** conditions. On the other hand, it is important to em- phasize that **conformal** **boundary** conditions for logarithmic CFTs are not fully classified: it might well be that there are more yet to be discovered, which would be related to bulk LCFTs by braid translation. This possibility and its potential consequences are discussed in more detail in the conclusion section.

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6.4. A **BOUNDARY** RG FLOW 131 and α = − cos γ 1 an attractive fixed point. Some numerical results show this RG flow explicitly in Figures 6.12 and 6.13 for the **loop** model and RSOS model respectively. To understand these figures one must first recall the discussion of finite size scaling for **models** described by Hamiltonians in section 5.3 and in particular equation (5.51). In both figures we have L 1 on the x-axis and on the y-axis we have the first gap, i.e. h 1 − h 0 , where h 1 and h 0 are the numerical approximations to the **conformal** dimensions appearing on the right hand side of (5.51) for the first excited state and the ground state respectively of the Hamiltonian. In Figure (6.12) the black curve corresponds to α = 0 and, as expected from the discussion in section 6.3.1, the gap converges logarithmically to zero. The red curve corresponds to α = − cos γ 1 and, as expected from the discussion in section 5.4.1, this gap to converges nicely to 1. Now let’s take the blue curve which corresponds to a slight perturbation away from α = 0. We observe that, for low sizes, the numerical approximation to h 1 − h 0 is very close to the result for α = 0, but that in the limit L 1 → 0 the curve converges towards 1, thus showing that α = 0 is a repulsive fixed point and that α = − cos γ 1 is an attractive fixed point - the interpretation is that of a lattice realisation of an RG flow away from a non-compact **boundary** **conformal** field theory towards a compact one. The same analysis for the RSOS model is presented in Figure 6.13.

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point function of operators with charge ˆ α 1 , which ex-
plains in what sense a continuum of ˆ α 1 charges appears
in the fusion of charges ˆ α 2 and ˆ α 3 .
The **loop** model also gives natural explanations to a well-known paradox encountered in trying to make c ≤ 1 Liouville into a consistent CFT. We see here that the vertex operator with ˆ α = 0 must be interpreted not as the identity but as a ‘marking’ operator, a feature very similar to what happens in the SLE construction. For in- stance, in the CFT proof [27] of Schramm’s left-passage formula [28] a point-marking (or “indicator” [27]) op- erator of zero **conformal** weight—but distinct from the identity operator—is used to select the correct **conformal** block in the corresponding correlation function. Similar arguments can be made within **boundary** CFT [29].

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The X-ray tomography is being used in the area of metallic foams to investigate micro-structure and determine pore size and similar geometric parameters [12, 13]. However, not so many studies involve direct flow simulation at pore scale level of open-cell metallic foams. Vicente et al. [14] used 3D tomographic data to determine physical properties of nickel–chromium metallic foams. They proposed morphological analysis methods that allow quantitative measures of the foam struc- ture. Furthermore, they conducted conductive heat transfer inside the metallic structure of the foam. Petrasch et al. [15] analyzed the permeability of a 10-ppi (pores per inch) reticulated porous ceramic from the Stokes flow regime to the unsteady flow regime. They reconstructed the 3D foam micro- structure geometry and conducted 3D incompressible fluid flow simulations for Reynolds number from 0.2 up to 200. Flow simulations were conducted using a finite volume method (ANSYS- CFX) with unstructured, body-fitted, tetrahedral mesh discretization. They compared the predicted permeability coefficients to the values determined by selected porous media flow **models**. The pro- posed method achieved good agreement with most of the other flow **models**. Magnico [16] studied hydrodynamic properties of metallic foams at the pore scale on a microtomographied sample from creeping flow to unsteady inertial flow. Both finite volume and Lattice Boltzmann methods were used and validated against experimental data. They solved the incompressible Navier–Stokes equa- tions using a second order, under-relaxed, SIMPLE finite volume formulation. No-slip **boundary** conditions were applied at the fluid–solid interface, and a prescribed pressure drop was imposed on a periodic computational domain. The method was validated against known solutions. In the work of Calvo et al. [17], X-ray micro-tomography and X-ray radiography were used to investi- gate single and two-phase flows in Ni-Cr metallic foams. They measured the pressure drop through Racemat RCM-NCX-1116 metallic foam for air velocities from 0.02 to 1.34 m=s. Pore scale pre- dictions were conducted using the Lattice Boltzman method [18] and compared with experiments as well as with other **models** proposed in the literature. Good agreement was achieved for the flow regimes tested.

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Introduction. The understanding of critical phenom- ena is now strongly intertwined with the study of the rich behavior of the q-state Potts model [ 1 ]. Aside from the historical spin representation [ 2 , 3 ], two other representa- tions of the Potts model have played a central role: the q- flow representation [ 4 , 5 ], which is a generalization of the **loop** description, and the Fortuin-Kasteleyn (FK) bond representation [ 6 , 7 ], which is also known as the random- cluster (RC) model. On one hand, theoretical advances were achieved thanks to the geometric and probabilis- tic interpretations they brought, as well as the extension to positive real q values [ 8 – 10 ]. For instance, they play an important role in **conformal** field theory [ 11 ] and in stochastic Loewner evolution [ 12 – 16 ]. On the other hand, numerical Monte Carlo (MC) methods, decisive in the study of not-exactly soluble **models**, have significantly benefitted from these insights. Indeed, the Metropolis [ 17 ] or heat-bath schemes rely on single-spin moves and often suffer from severe critical slowing-down [ 18 , 19 ], and the Sweeny algorithm [ 20 ], a local-bond update scheme, has complications from connectivity-checking. Based on the coupling between spin and FK represen- tations [ 6 , 7 , 21 ], efficient cluster methods, including the Swendsen-Wang (SW) and Wolff algorithms [ 22 , 23 ], have been developed and widely used. For the q-flow repre- sentation, one can apply the Prokof’ev-Svistunov worm algorithm [ 24 – 27 ], which has proven to be particularly ef- ficient at computing the magnetic susceptibility [ 28 ] and the spin-spin correlation function [ 29 ].

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I. SATURATION FROM QCD DIPOLES
A regime of QCD at small coupling constant where the density of partons begins to saturate has since long been the subject of many studies [1–18]. In one class of **models**, one expects the effect due to the high energy multiplication of partons due to the QCD dynamics. Another way which was proposed to investigate high density partonic effects is the consideration of collisions at high energy on heavy nuclei. Indeed, the number of partons is supposed to be large in the background of the collision providing **boundary** conditions favorable for the saturation mechanism to happen. In the present paper, we will focus on the first scheme, i.e. the saturation mechanism due to the energy evolution of the parton density in a purely perturbative QCD framework. More precisely, we will consider even “perturbative QCD” **boundary** conditions, in order to select the features of saturation which could be fully calculated from the QCD lagrangian.

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By conformal invariance, we can define the level line of the GFF in any simply connected domain starting from a boundary point and targeted at another boundary point.. This pro[r]

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We have now shown that c 0 > 0, but we still have to show that c 0 < ∞. We use a
similar coupling with the fractal percolation model. For any dyadic square that does not touch the **boundary** of [0, 1] 2 , we let X(C) be 0 if C is surrounded by a **loop** in L that
is contained in the set of eight neighboring dyadic squares to C (of the same size). The X(C) are i.i.d. (for all C whose eight neighbors are contained in D), and have a small probability (say smaller than 1/4) of being 1 when c is taken sufficiently large. We now use the fact, mentioned above, that if p ≤ 1/4, then M is almost surely empty; from this we conclude easily that almost surely, for each C (whose incident neighbors are in D) every point in C is surrounded by a **loop** in L almost surely. It follows immediately that almost surely, every point in D is surrounded by a **loop** in L. This implies that almost surely all loops of L belong to the same cluster (since otherwise there would be a point on the **boundary** of a cluster that was not surrounded by a **loop**).

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Making a mathematical theory of the BPZ approach triggered in the 80’s and 90’s intense research in the field of Vertex Operator Algebras (VOA for short) introduced by Borcherds [ Bo86 ] and Frenkel-Lepowsky- Meurman [ FLM89 ] (see also the book [ Hu97 ] and the article [ HuKo07 ] for more recent developments on this formalism). Even if the theory of VOA was quite successful to rigorously formalize numerous CFTs, the approach suffers certain limitations at the moment. First, correlations are defined as formal power series (convergence issues are not tackled in the first place and are often difficult); second, many fundamental CFTs have still not been formalized within this approach, among which the CFTs with uncountable collections of primary fields and in particular Liouville **conformal** field theory (LCFT in short) studied in this paper. Moreover, the theory of VOA, which is based on axiomatically implementing the operator product expansion point of view of physics, does not elucidate the link to the the path integral approach or to the **models** of statistical physics at critical temperature (if any).

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Other pieces of Ψ g localized near Φ i α = 0, i.e. the (quantum) Coulomb branch of the
theory, have also been studied [ 21 ]. Unlike the Higgs branch quiver with a closed **loop**, superpotential and an exponential growth in its number of ground states, it was found that the number of Coulomb branch ground states grows only polynomially in the κ i . Interpreting the q i as the relative positions of wrapped branes, these ground states can be viewed as describing various multiparticle configurations, as we will soon proceed to describe in further detail. To each ground state in the quantum Coulomb branch there exists a corresponding ground state in the Higgs branch, but the converse is not true. Another way to view this statement is that whenever a given Ψ g has non-trivial structure

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viii
This expectation is quite natural if we take a look at the analogous symmetric situation: suppose that L is a holomorphic line bundle over (M, J) and that there exists a non-degenerate holomorphic section g ∈ H 0 (M, S 2 T ∗ M ⊗ L), called a holomorphic **conformal** structure. Still under the Kählerness assumption, Inoue, Kobayashi and Ochiai [IKO80] proved that if L is holomorphically trivial then (M, J) is a ﬁnite quotient of a complex torus. On the other hand, if L is not trivial, then new examples appear, and in fact classiﬁcations are known only for the compact surfaces ([KO82]), and for projective threefolds ([JR05]). Let us note that the standard example of such a manifold is given by the hyperquadric:

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gregory.korchemsky@cea.fr , snegro@insti.physics.sunysb.edu , grisha.sizov@gmail.com
Abstract: We study integrability of fishnet-type Feynman graphs arising in planar four- dimensional bi-scalar chiral theory recently proposed in arXiv:1512.06704 as a special dou- ble scaling limit of gamma-deformed N = 4 SYM theory. We show that the transfer matrix “building” the fishnet graphs emerges from the R−matrix of non-compact confor- mal SU(2, 2) Heisenberg spin chain with spins belonging to principal series representations of the four-dimensional **conformal** group. We demonstrate explicitly a relationship between this integrable spin chain and the Quantum Spectral Curve (QSC) of N = 4 SYM. Using QSC and spin chain methods, we construct Baxter equation for Q−functions of the con- formal spin chain needed for computation of the anomalous dimensions of operators of the type tr(φ J

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Acknowledgements. C. Guillarmou akcnowledges that this project has received funding from the Euro- pean Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme, grant agreement No 725967. A. Kupiainen is supported by the Academy of Finland and ERC Advanced Grant 741487. R. Rhodes is partially supported by the Institut Universitaire de France (IUF). The authors wish to thank Zhen-Qing Chen, Naotaka Kajino for discussions on Dirichlet forms, Ctirad Klimcik and Yi- Zhi Huang for explaining the links with Vertex Operator Algebras, Slava Rychkov for fruitful discussions on the **conformal** bootstrap approach, Alex Strohmaier, Tanya Christiansen and Jan Derezinski for discussions on the scattering part and Baptiste Cercle for comments on earlier versions of this manuscript.

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Chapter II : Sequential **models** of digital tracking loops
II.2 Code tracking with Delay Lock Loops 57
II.2 Code tracking with Delay Lock Loops
A Delay Lock **Loop** is a feedback system that is able to track the phase of a pseudo-random noise (PRN) signal. A DLL is able to synchronize its own local PRN replica with the incoming PRN signal, so that pseudorange measurements can be derived from measurements of the local code phase register. The architecture of a DLL is shown on Figure II-7. A DLL is based on correlation properties of PRN codes so that correlation measurements between received signal and local replicas have to be performed in order to provide a synchronization error signal to the feedback process. This error signal is then filtered by the **loop** filter to provide a control signal to correct the phase of the local code. Expression of the noiseless received signal is

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