8 Conclusions
We have analyzed the full counting statistics of the transverse and staggered magnetization of a subsystem in the thermodynamic **limit** of the transverse field **Ising** **chain**. We derived a con- venient determinant representation for the corresponding generating functions χ (u,s) (λ, `, t). We first considered the FCS in equilibrium states and showed that the probability distributions are always non-Gaussian except in the **limit** of infinite subsystem size at finite temperature. We determined the temperature and field dependence of the generating function as well as the first few cumulants. We then moved on to the main focus of our work, the calculation of the FCS after quantum quenches. We considered two quench protocols: transverse field quenches and evolution starting from a classical Néel state. We first determined the FCS in the stationary states reached at late times. The probability distributions are again non Gaus- sian, except in the **limit** of infinite subsystem size. We analyzed the time evolution of the probability distributions P (u,s) (m, t) for a variety of quenches by numerically evaluating the exact determinant representation for the generating function (the numerical errors incurred are negligible). For transverse field quenches originating in the paramagnetic phase P (u,s) (m, t) showed interesting smoothing and broadening behaviour in time. In contrast, P (u,s) (m, t) dis- played a simpler behaviour for quenches originating in the ferromagnetic phase. In the case of a Néel quench P (s) (m, t) encoded detailed information on the restoration of translational invariance. The numerical approach provided us with evidence for the existence of a scal- ing regime for the generating function in which we observed data collapse according to the **scaling** form (59). This is turn allowed us to proceed with the derivation of the main result of our work: the analytic expression (100) for the FCS after transverse field quenches in the space-time **scaling** **limit** t , ` → ∞, t/` fixed. This was achieved by a substantial generalization of the multi-dimensional stationary phase approximation method of Refs [ 42 , 46 ]. We per-

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In this work we begin with a presentation of the **scaling** theory for the density profiles in the presence of gradient field inhomogeneities. Specifically, we consider a system with a deviation from the critical coupling which varies linearly in one space direction. The coupling is at **its** critical value in the middle of the system where an interface region separates the ordered phase on the left from the disordered phase on the right. We test the validity of the **scaling** arguments, first at the mean-field level, within Ginsburg-Landau theory. Then we present a study of the **Ising** quantum **chain** in a linearly varying transverse field, h l = 1 + gl, which corresponds to the extreme anisotropic **limit** of the two-dimensional classical **Ising** model with a linear variation of the couplings. We work in the **scaling** **limit** where the size L of the system goes to infinity while the gradient g goes to zero with the product gL held fixed. The excitation spectrum of the inhomogeneous **Ising** quantum **chain** is obtained exactly in terms of the solution of an harmonic oscillator eigenvalue problem. The knowledge of the eigenvectors allows us to obtain the energy density profile, the magnetization profile and the behaviour of the spin-spin correlation function. Their **scaling** forms are in complete agreement with the results of the **scaling** theory.

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In particular, the annealed pressure obtained in Proposition 7.3 is so complicated that we can not even prove **its** differentiability. Without the differentiability, we cannot go further to the other thermodynamic quantities or **limit** theorems. It is worth noting that in [17], the authors conjecture that when the degrees of vertices fluctuate, the annealed and quenched models behave differently. In particular, they guess that the critical temperatures in these two models are not equal.

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The model described in the paper is based on comparison of two opposite client characteristics, one leading to potential credit **limit** increase, and the second leading to decrease of the credit **limit**. Credit **limit** calculation involved client 's personal data, which allows to approach each client on the individual basis during credit amount allocation. The **scaling** method was applied in order to analyze the data obtained from the real client database. The **scaling** ratio provides reasonable predictive capability from the risk point of view and therefore has been proposed to serve as a model for credit **limit** allocation. The model's flexibility allows coefficients' adjustments according to new statistical data. Although the present work is quite preliminary, it does indicate that presented solution allows to substantially increase the credit portfolio while maintaining **its** quality.

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Several works have used this diagnostic to argue for the existence of quantum butterfly effect 7–10 and extract the Lyapunov exponent, with examples including the O(N ) model, 11 fermionic models with critical Fermi surface, 12 and weakly diffusive metals. 13 On the other hand, some systems, for example Luttinger liquids 14 and many-body localized systems, 15–19 do not show the Lyapunov behav- ior and are hence characterized as less chaotic or as slow scramblers. Also, some works have shown that in cer- tain Hamiltonians, the exponent extracted from OTOC does not match the classical counterpart of the semiclas- sical **limit**. 20,21 For systems with bounded local Hilbert space and Hamiltonians with local interactions, a work 22 proposed that the density-OTOC is a more suitable di- agnostic.

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In this proof, we shall restrict to the quartic model described earlier and use a univer- sality argument, presented in the next section, to extend the proof to the generic model.
Our main reason for doing so is that the universality argument is succinct yet powerful.
As one will see, the following analysis is quite involved and tailored for the quartic model only. To prove these results for generic models directly is an arduous task, deserving **its** own paper.

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In this paper we will be particularly, but not exclusively, interested in the specific case where the underlying field h is a 2d Gaussian free field with zero-boundary conditions. In this case the measure µ γ (when it is defined and non-zero) is known
as the Liouville measure with parameter γ. This has been an object of considerable recent interest due to **its** strong connection with 2d Liouville quantum gravity and the KPZ relations [DS11, RV11, Ber15b]. Recent works in the case γ < 2 include [DS11, RV11, Ber15a], which among other things make an in-depth study of **its** moments, multifractal structure, and universality. Recently, in [APS17], it has also been shown that these measures can be approximated using so-called local sets of the Gaussian free field. This is a particularly natural construction because it is both local and conformally invariant.

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exact representations of fractional lattice Laplacian defined on finite periodic linear chains and to the analysis of **its** continuum limits. An overview on the relations between fractional calculus and fractal curves such as the Weierstrass Mandelbrot function has been presented by West [29].
Power law behavior which is naturally described by fractional calculus occurs in various completely differ- ent contexts such as anomalous and turbulent diffusion, critical phenomena such as phase transitions, biological systems, the human economy, and last but not least the present crisis of the financial system teaches us that complex systems such as the world economy does not obey gaussian statistics where extreme events are ex- tremely seldom, however they are governed by stable heavy tailed L´evy distributions where extreme events are in the heavy tail still rare, but they are anyway much more likely as in gaussian cases.

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X
i=1
(σ z i + λσ y i σ y i+1 − iκσ y i ). (2.2) **Its** eigenvalues and those of H(λ, κ) are therefore either all real or occur in complex conju- gate pairs. This is precisely the well known behaviour one finds when H(λ, κ) is symmetric with respect to an anti-linear operator [12, 13, 14, 15, 16], which as mentioned above has recently attracted a lot of attention. In quantum mechanical or field theoretical models the anti-linear operator is commonly taken to be the PT -operator, which carries out a simul- taneous parity transformation P : x → − x and time reversal T : t → − t. When acting on complex valued functions the anti-linear operator T is understood to act as complex con- jugation. Real eigenvalues are then found for unbroken PT -symmetry, meaning that both the Hamiltonian and the eigenfunctions remain invariant under PT -symmetry, whereas broken PT -symmetry leads to complex conjugate pairs of eigenvalues.

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This has opened up opportunities for the country’s telcos. For example, MTN is playing an important role in developing services in the three main categories of mobile health services outlined by the government: prevention, health system strengthening and health worker empowerment. Etisalat is also active in developing innovative health services. For example **its** Mobile Baby application has trained more than 500 birth attendants and midwives and registered more than 10,000 pregnant women in the programme. Sproxil’s SMS-based drug verification system, launched in 2010, has also benefited from government support.

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Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, 50923 Ko¨ln, Germany and NIC c/o Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
~Received 30 November 1998!
We consider the paramagnetic phase of the random transverse-field **Ising** spin **chain** and study the dynamical properties by numerical methods and **scaling** considerations. We extend our previous work @Phys. Rev. B 57, 11 404 ~1998!# to new quantities, such as the nonlinear susceptibility, higher excitations, and the energy- density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of @(J/h) 1/z # av 51. Particularly, whereas the average spin autocorrelation function in imaginary time decays as @G# av ( t);t 21/z , the average energy-density autocorrelations decay with another exponent as @G e # av ( t);t 2221/z .

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We call the region T cs < ≤ T T cf as renormalization region, rather than merely the criticality one.
The **scaling** laws are based on the **scaling** hypothesis, which is purely a con- jecture. Taking **Ising** model as an example, we interpret it by Euclidean geometry and fractal geometry. From the point of view of Euclidean geometry, the reason why the GR theory works is that the lattice correlation length becomes infinity in the vicinity of T cf , such that the influence of all finite microscopic characteristic lengths are wiped out, and the correlation length is a unique characteristic quan- tity. The singularities of thermodynamic parameters are attributed to the corre- lation length singularity, and any size transformation can’t change the form of the free energy function, only changes the parameter **scaling**. On the other hand, the transformation hierarchy is emphasized in terms of fractal geometry: The formation of blocks with finite side length n begins at T cf , they can go through r-hierarchical SS transformations at the same temperature, and the original lat- tice spin system is replaced by the block spin system. For any finite value of r , one can find the same form of the free energy function represented by the block spins, no matter what value of r will be. Clearly, both interpretations describe the same phenomena. It’s impossible to explain in terms of Euclidean geometry the uniqueness of the block side length n * at the critical point and the coexistence of n + -blocks and n − -blocks. In fact, the renormalization transformation is just the self similar one. In a word, the geometrical structure and the physical para- meters function forms on the (r + 1)-th hierarchy maintain the original ones on the r-hierarchy. By the forms, we are unable to recognize which belong to the r-hierarchy and what is the actual value of r. The indistinguishability means symmetry [7]. The SS transformation comes essentially from the spacetime ho- mogeneity and is a special type of symmetrical transformation, differing from the conventional operations such as rotation, translation, inversion, etc. The GR is a symmetry group, **its** corresponding conserved quantity is the **scaling**, and there exists **scaling** invariance. The **scaling** laws do be the conservation laws.

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1/4 ψ ˙ (k+1) n ∗ LM (k+1) n
6 k · 12 n n k−5/4 n→∞ −→ Υ ˙ (k+1) CLM ˙ (k+1) 1 .
The proof follows exactly the same lines as Proposition 25, so we leave it to the reader.
One has to be a little careful about the small variations in the construction of Section 5.4, compared to Proposition 23, when we are dealing with the distinguished edge and **its** adjacent edges, but these variations disappear in the **scaling** **limit**. Also, note that the reason why the measure dσ σ k−5/4 appears in the disintegration of CLM ˙ (k+1) , instead of dσ σ k−9/4 , is that CLM ˙ (k+1) carries intrinsically the location of a distinguished point (corresponding to the only vertex of the scheme that has degree 1). This marked point should be seen as the continuum counterpart of the root edge in discrete maps, so it is natural that if the continuum object has a total “mass” σ, then the marking introduces a further factor σ.

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NELLY LITVAK AND PHILIPPE ROBERT
Abstract. Motivated by an original on-line page-ranking algorithm, starting from an arbitrary Markov **chain** (C n ) on a discrete state space S, a Markov **chain** (C n , M n ) on the product space S 2 , the cat and mouse Markov **chain**, is constructed. The first coordinate of this Markov **chain** behaves like the original Markov **chain** and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov **chain** are investigated. A representation of **its** invariant measure is in particular obtained. When the state space is infinite it is shown that this Markov **chain** is in fact null recurrent if the initial Markov **chain** (C n ) is positive recurrent and reversible. In this context, the **scaling** properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in Z and Z 2 , reflected simple random walk in N and also in a continuous time setting. For several of these processes, a time **scaling** with rapid growth gives an interesting asymptotic behavior related to **limit** results for occupation times and rare events of Markov processes.

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Meanwhile, the contribution of the private sector is vital.
Through industry bodies have an important role to play in helping to draw up standards and codes.
At the same time, the public sector needs to ensure that **its** workforce has the skills needed for a transformed CVC, and it has a vital role in raising awareness of the value proposition in innovative CVC technologies and services – through case studies, demonstrations and consumer outreach.

Dennard **scaling** came to an end in 2005 with the development of 90nm lithography. At this level, transistor gates become too thin to prevent current from leaking into the substrate, resulting in a rise in power density. The well-known relationship 𝑃 𝑑 = 𝐶 𝑉 2 𝑓 illustrates the relationship between clock speed and power consumption where 𝑃 𝑑 is the transistor dynamic switching power consumption; 𝐶 is the CMOS switch lumped capacitance which is the sum of the junction capacitances and gate capacitances; 𝑉 is the supply voltage; and 𝑓 is the clock frequency. The above expression for 𝑃 𝑑 can be justified by considering a CMOS inverter for example. Here a 1-0 transition charges the equivalent capacitance through the source-drain of the PMOS type transistor, dissipating half the energy and in the 0-1 transition the capacitor dumps the stored charge through the source-drain of the NMOS transistor resulting in an approximate total energy dissipation of 2 ∗ 1 2 𝐶 𝑉 2 over one cycle.

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The CIM achieves similar success probabilities for cubic and dense graphs, suggesting that dense problems are not intrinsically harder than sparse ones for this class of annealer. D-Wave’s strong dependence on edge density is most likely a consequence of embedding compactness: it is known that more compact embeddings (fewer physical qubits per **chain**) tend to give better annealing performance, after all optimization and parameter setting is considered [20]. Since qubits on the D-Wave chimera graph have at most 6 connections, the minimum **chain** length is ` = d(d − 2)/4e, so embeddings grow less compact with increasing graph degree (see Supp. Sec. S3). Since degree-1 and degree-2 vertices can be pruned from a graph in polynomial time (a variant of cut-set conditioning [44]), d = 3 is the minimum degree required for NP-hardness. Of NP-hard MAX-CUT instances, Fig. 3(c) suggests that there is only a very narrow region (d = 3, 4) where D-Wave matches or outperforms the CIM; for the remainder of the graphs the CIM dominates.

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As is common for weak limits of light-tailed Metropolis–Hasting algorithms, the limits of Metropolis–Hastings for Curie–Weiss are simple univariate diffusion processes. Interestingly, in determining the **scaling** **limit** of Lifted Metropolis–
Hastings we obtain an elementary Markov process which has so far received only very limited attention in the literature. The **limit** process is a one-dimensional piecewise deterministic Markov process which we will refer to as a zig-zag pro- cess: the process moves at a deterministic and constant speed, until it switches direction and moves at the same speed but in the opposite direction. The switch- ing occurs at a time-inhomogeneous rate which is directly related to the derivative of the density function of **its** stationary distribution. We analyse this zig-zag pro- cess in some detail, establishing in particular exponential ergodicity under mild conditions.

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Although we will not apply it in **its** following form, for completeness, it seems appro- priate to put all the above pieces together and state a (simpliﬁed version of) the main result of [1], which demonstrates the convergence of the rescaled C 1 n . So as to make C 1 n
into a metric space, it is assumed to be equipped with the usual shortest path graph distance d C n 1 . Note that the convergence of the ﬁrst coordinate was originally proved as part of [4], Corollary 2.

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Abstract
We consider the double **scaling** **limit** for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t ∗ . In a previous paper, the **scaling** limits for the positions of the paths at time t 6= t ∗ were shown to be the usual **scaling** limits from random matrix theory. Here, we describe the **limit** as n → ∞ of the correlation kernel at critical time t ∗ and in the double **scaling** regime. We derive an integral representation for the **limit** kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and **its** matching with a global parametrix.

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