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Electromagnetic microtearing modes (MTMs) have been observed in many different spherical tokamak regimes. Understanding how these and other electromagnetic modes nonlinearly saturate is likely critical in understanding the confinement of a high $\beta$ spherical tokamak (ST). Equilibrium $E\times B$ sheared flows have sometimes been found to significantly suppress low $\beta$ ion scale transport in both gyrokinetic simulations and in experiment. This work aims to understand the conditions under which $E\times B$ sheared flow impacts on the saturation of MTM simulations. Two experimental regimes are examined from MAST and NSTX, on surfaces that have unstable MTMs. The MTM driven transport on a local flux surface in MAST is shown to be more resilient to suppression via $E\times B$ shear, compared to the case from NSTX where the MTM transport is found to be significantly suppressed. This difference in the response to flow shear is explained through the impact of magnetic shear, $\hat{s}$ on the MTM linear growth rate dependence on ballooning angle, $\theta_0$. At low $\hat{s}$, the growth rate depends weakly on $\theta_0$, but at higher $\hat{s}$, the MTM growth rate peaks at $\theta_0 = 0$, with regions of stability at higher $\theta_0$. Equilibrium $E\times B$ sheared flows act to advect the $\theta_0$ of a mode in time, providing a mechanism which suppresses the transport from these modes when they become stable. The dependence of $\gamma^{MTM}$ on $\theta_0$ is in qualitative agreement with a recent theory [M.R. Hardman et al (2023)] at low $\beta$ when $q\sim1$, but the agreement worsens at higher $q$ where the theory breaks down. This work highlights the important role of the safety factor profile in determining the impact of equilibrium $E\times B$ shear on the saturation level of MTM turbulence., Comment: 20 pages, 16 figures

Quantum Key Distribution (QKD) promises to revolutionize the field of security in communication, with applications ranging from state secrets to personal data, making it a key player in the ongoing battle against cyber threats. Reference-Frame-Independent (RFI) QKD aims to simplify QKD implementations by allowing to reduce the requirements of alignment on a shared reference frame. This is done by performing two mutually unbiased measurements on the control states. In this work, we present a novel fully passive receiver for time-bin encoded RFI-QKD. Conversion of time-bin to polarization is employed to perform the required quantum measurement in a fully passive manner. Furthermore, to overcome experimental errors, we retrieved a complete description of our measurement apparatus by employing a recently introduced Quantum Detector Self-Characterization technique, without performing tomographic studies on the detection stage. In fact, the security analysis carried out in this work uses experimentally retrieved Positive Operator Valued Measurements, which consider our receiver defects, substituting the ideal expected operators and thus increasing the overall level of secrecy. Lastly, we conducted a proof-of-principle experiment that validated the feasibility of our method and its applicability to QKD applications., Comment: 13 pages, 6 figures, 1 table

Plasma turbulence on disparate spatial and temporal scales plays a key role in defining the level of confinement achievable in tokamaks, with the development of reduced numerical models for cross-scale turbulence effects informed by experimental measurements an essential step. MAST-U is a well-equipped facility having instruments to measure ion and electron scale turbulence at the plasma edge. However, measurement of core electron scale turbulence is challenging, especially in H mode. Using a novel synthetic diagnostic approach, we present simulated measurement specifications of a proposed mm-wave based collective scattering instrument optimised for measuring both normal and binormal electron scale turbulence in the core and edge of MAST-U. A powerful modelling framework has been developed that combines beam-tracing techniques with gyrokinetic simulations to predict the sensitivity, localisation and spectral range of measurement. For the reconstructed MAST 022769 shot, a maximum measurable normalised bi-normal wavenumber of $k_{\perp} \rho_{e} \sim 0.6$ was predicted in the core and $k_{\perp} \rho_{e} \sim 0.79$ near the pedestal, with localisation lengths $L_{FWHM}$ ranging from $\sim$ 0.4 m in the core at $k_{\perp} \rho_{e} \sim 0.1$ to ~0.08m at $k_{\perp} \rho_{e} \sim 0.45$. Synthetic diagnostic analysis for the 022769 shot using CGYRO gyrokinetic simulation spectra reveal that ETG turbulence wavenumbers of peak spectral intensity comfortably fall within the measurable range of the instrument from the core to the pedestal. The proposed diagnostic opens up opportunities to study new regimes of turbulence and confinement in association with upcoming non-inductive, microwave based current drive experiments on MAST-U and can provide insight into cross-scale turbulence effects, while having suitability to operate during burning plasma scenarios on future reactors such as STEP., Comment: 22 pages, 19 figures, journal submission

We prove that in strongly disordered, interacting, quantum chains, the conductance of a chain of length $L$ vanishes faster than $1/L$. This means that transport is anomalous in such chains. This phenomenon was first claimed in \cite{Basko2006,gornyi2005interacting} and a pioneering treatment appeared in \cite{Imbrie2016a}, Comment: 71 pages

The modelling of symmetric rigid dumbbell particles suspended in a Newtonian fluid, as a model of a rigid-rod polymeric solution, has been accomplished exclusively through the diffusion equation, which has been detailed elegantly by Bird et al. [Chapter 14 of $Dynamics \ of \ Polymeric \ Liquids$, Vol 2, Ed 2, (1987)]. In this tutorial, a straightforward approach for modelling a lopsided rigid dumbbell particle is presented by the Langevin analysis. The connector force between the dumbbell beads is obtained through the rigidity constraint of the center-to-center vector of the dumbbell using its Langevin equation. By directly averaging via the Langevin equation, the evolution of the center-to-center vector, and the configuration tensor are derived. The stress expressions for the dumbbell from the Langevin equation, and the diffusion equation for the orientation distribution function of the center-to-center vector of the dumbbell are also derived, and the final expressions agree with established results from other methods.

We revisit and expand the analysis in [15] of the localized phase of disordered pinning models. The arguments are developed for i.i.d. site disorder on which we assume only that the moment generating function is bounded in a neighborhood of the origin. Quantitative $C^\infty$ estimates on the free energy density are established, showing in particular that the regularity class is at least Gevrey-3. After explaining how a quenched concentration bound and the quenched Central Limit Theorem (CLT) on the number of the pinned sites, i.e., the contact number, can be extracted from the regularity estimates on the free energy, we establish a quenched Local CLT for the same quantity. The centering in these CLTs is random in the sense that it is disorder dependent and a concentration bound and the CLT are established also for the centering sequence, as well as a Hardy-Littlewood random walk type-estimate for its fluctuations.

We derive the constitutive equation for a suspension of lopsided Hookean dumbbells. By lopsided, we mean that one bead is larger than the other. We find that all results derived for symmetric Hookean dumbbells can be taken over for lopsided ones by replacing$ \frac{2}{\zeta}$ with $\frac{1}{\zeta_1}+\frac{1}{\zeta_2}$, where ${\zeta}$, ${\zeta}_1$ and ${\zeta}_2$ are the bead friction coefficients for the symmetric dumbbell beads, and the first and second beads of the lopsided dumbbell respectively.

Starting from the assumption that saturation of plasma turbulence driven by temperature-gradient instabilities in fusion plasmas is achieved by a local energy cascade between a long-wavelength outer scale, where energy is injected into the fluctuations, and a small-wavelength dissipation scale, where fluctuation energy is thermalized by particle collisions, we formulate a detailed phenomenological theory for the influence of perpendicular flow shear on magnetized-plasma turbulence. Our theory introduces two distinct regimes, called the weak-shear and strong-shear regimes, each with its own set of scaling laws for the scale and amplitude of the fluctuations and for the level of turbulent heat transport. We discover that the ratio of the typical radial and poloidal wavenumbers of the fluctuations (i.e., their aspect ratio) at the outer scale plays a central role in determining the dependence of the turbulent transport on the imposed flow shear. Our theoretical predictions are found to be in excellent agreement with numerical simulations of two paradigmatic models of fusion-relevant plasma turbulence: (i) an electrostatic fluid model of slab electron-scale turbulence, and (ii) Cyclone-base-case gyrokinetic ion-scale turbulence. Additionally, our theory envisions a potential mechanism for the suppression of electron-scale turbulence by perpendicular ion-scale flows based on the role of the aforementioned aspect ratio of the electron-scale fluctuations., Comment: 21 pages, 13 figures, 1 table

A quasi-linear reduced transport model is developed from a database of high-$\beta$ electromagnetic nonlinear gyrokinetic simulations performed with Spherical Tokamak for Energy Production (STEP) relevant parameters. The quasi-linear model is fully electromagnetic and accounts for the effect of equilibrium flow shear using a novel approach. Its flux predictions are shown to agree quantitatively with predictions from local nonlinear gyrokinetic simulations across a broad range of STEP-relevant local equilibria. This reduced transport model is implemented in the T3D transport solver that is used to perform the first flux-driven simulations for STEP to account for transport from hybrid-KBM turbulence, which dominates over a wide region of the core plasma. Nonlinear gyrokinetic simulations of the final transport steady state from T3D return turbulent fluxes that are consistent with the reduced model, indicating that the quasi-linear model may also be appropriate for describing the transport steady state. Within the assumption considered here, our simulations support the existence of a transport steady state in STEP with a fusion power comparable to that in the burning flat-top of the conceptual design, but do not demonstrate how this state can be accessed.

[ABRIDGED] This paper discusses the importance of parallel perturbations of the magnetic-field in gyrokinetic simulations of electromagnetic instabilities and turbulence at mid-radius in the burning plasma phase of the conceptual high-$\beta$, reactor-scale, tight-aspect-ratio tokamak STEP. Previous studies have revealed the presence of unstable hybrid kinetic ballooning modes (hKBMs) at binormal scales approaching the ion Larmor radius. In this STEP plasma it was found that the hKBM requires the inclusion of parallel magnetic-field perturbations to be linearly unstable. Here, the extent to which the inclusion of fluctuations in the parallel magnetic-field can be relaxed is explored through gyrokinetic simulations. In particular, the frequently used MHD approximation (dropping $\delta \! B_{\parallel}$ and setting the $\nabla B$ drift frequency equal to the curvature drift frequency) is discussed and simulations explore whether this approximation is useful for modelling STEP plasmas. It is shown that the MHD approximation can reproduce some of the linear properties of the full STEP gyrokinetic system, but is too stable at low $k_y$ and nonlinear simulations using the MHD approximation result in very different transport states. It is demonstrated that the MHD approximation is challenged by the high $\beta^{\prime}$ values in STEP, and that the approximation improves considerably at lower $\beta^{\prime}$. Furthermore, it is shown that the sensitivity of STEP to $\delta \! B_{\parallel}$ fluctuations is primarily because the plasma sits close to marginality and it is shown that in slightly more strongly driven conditions the hKBM is unstable without $\delta \! B_{\parallel}.$ Crucially, it is demonstrated that the state of large transport typically predicted by local electromagnetic gyrokinetic simulations of STEP plasmas is not solely due to $\delta \! B_{\parallel}$ physics., Comment: 29th IAEA FEC conference

We consider the scaling-invariant nonlocal Willmore energy, defined via the nonlocal mean curvature by Caffarelli, Roquejoffre and Savin. Our main result is the existence of minimizers in the class of convex $C^1$-curves.

We study the configurations of the nearest neighbor Ising ferromagnetic chain with IID centered and square integrable external random field in the limit in which the pairwise interaction tends to infinity. The available free energy estimates for this model show a strong form of disorder relevance, i.e., a strong effect of disorder on the free energy behavior, and our aim is to make explicit how the disorder affects the spin configurations. We give a quantitative estimate that shows that the infinite volume spin configurations are close to one explicit disorder dependent configuration when the interaction is large. Our results confirm predictions on this model obtained in D. S. Fisher, P. Le Doussal and C. Monthus (Phys. Rev. E 2001) by applying the renormalization group method introduced by D. S. Fisher (Phys. Rev. B 1995)., Comment: 37 pages, 7 figures. Changes wrt v1: added some references, 3 figures, and local changes

Integration of Quantum Key Distribution (QKD) in existing telecommunication infrastructure is crucial for the widespread adoption of this quantum technology, which offers the distillation of unconditionally secure keys between users. In this letter, we report a field trial between the Points of Presence (POPs) placed in Treviso and in Venezia - Mestre, Italy, exploiting the QuKy commercial polarization-based QKD platforms developed by ThinkQuantum srl and two different standards of single-mode optical fibers, i.e. G.652 and G.655, as a quantum channel. In this field trial, several configurations were tested, including the co-existence of classical and quantum signals over the same fiber, providing a direct comparison between the performances of the G.652 and G.655 fiber standards for QKD applications., Comment: 6 pages, 5 figures

We present herein the results of a linear gyrokinetic analysis of electromagnetic microinstabilites in the conceptual high-$\beta$, reactor-scale, tight-aspect-ratio tokamak STEP (Spherical Tokamak for Energy Production, https://step.ukaea.uk). We examine a range of flux surfaces between the deep core and the pedestal top for two candidate flat-top operating points of the prototype device. Local linear gyrokinetic analysis is performed to determine the type of microinstabilities that arise under these reactor-relevant conditions. We find that the equilibria are dominated at ion binormal scales by a hybrid version of the Kinetic Ballooning Mode (KBM) instability that has significant linear drive contributions from the ion temperature gradient and from trapped electrons, while collisional Microtearing Modes (MTMs) are sub-dominantly also unstable at similar binormal scales. The hybrid-KBM and MTM exhibit very different radial scales. We study the sensitivity of these instabilities to physics parameters, and discuss potential mechanisms for mitigating them. The results of this investigation are compared to a small set of similar conceptual reactor designs in the literature. A detailed benchmark of the linear results is performed using three gyrokinetic codes; alongside extensive resolution testing and sensitivity to numerical parameters providing confidence in the results of our calculations, and paving the way for detailed nonlinear studies in a companion article.

In this work, we present first-of-their-kind nonlinear local gyrokinetic simulations of electromagnetic turbulence at mid-radius in the burning plasma phase of the conceptual high-$\beta$, reactor-scale, tight-aspect-ratio tokamak STEP (Spherical Tokamak for Energy Production). A prior linear analysis in D. Kennedy et al. 2023 Nucl. Fusion 63 126061 reveals the presence of unstable hybrid kinetic ballooning modes, where inclusion of the compressional magnetic field fluctuation, $\delta B_{\parallel}$, is crucial, and subdominant microtearing modes are found at binormal scales approaching the ion-Larmor radius. Local nonlinear gyrokinetic simulations on the selected surface in the central core region suggest that hybrid kinetic ballooning modes can drive large turbulent transport, and that there is negligible turbulent transport from subdominant microtearing modes when hybrid kinetic ballooning modes are artificially suppressed (through the omission of $\delta B_{\parallel}$). Nonlinear simulations that include perpendicular equilibrium flow shear can saturate at lower fluxes that are more consistent with the available sources in STEP. This analysis suggests that hybrid kinetic ballooning modes could play an important role in setting the turbulent transport in STEP, and possible mechanisms to mitigate turbulent transport are discussed. Increasing the safety factor or the pressure gradient strongly reduces turbulent transport from hybrid kinetic ballooning modes in the cases considered here. Challenges of simulating electromagnetic turbulence in this high-$\beta$ regime are highlighted. In particular the observation of radially extended turbulent structures in the absence of equilibrium flow shear motivates future advanced global gyrokinetic simulations that include $\delta B_\parallel$.

We investigate surfaces with bounded L^p-norm of the fractional mean curvature, a quantity we shall refer to as fractional Willmore-type functional. In the subcritical case and under convexity assumptions we show how this Willmore-functional controls local parametrization, and conclude as consequences lower Ahlfors-regularity, a weak Michael-Simon type inequality, and an application to stability.

We consider the continuum version of the random field Ising model in one dimension: this model arises naturally as weak disorder scaling limit of the original Ising model. Like for the Ising model, a spin configuration is conveniently described as a sequence of spin domains with alternating signs (domain-wall structure). We show that for fixed centered external field and as spin-spin couplings become large, the domain-wall structure scales to a disorder dependent limit that coincides with the infinite disorder fixed point process introduced by D. S. Fisher in the context of zero temperature quantum Ising chains. In particular, our results establish a number of predictions that one can find in [Fisher, Le Doussal and Monthus 2001]. The infinite disorder fixed point process for centered external field is equivalently described in terms of the process of suitably selected extrema of a Brownian trajectory introduced and studied by J. Neveu and J. Pitman [Neveu and Pitman 1989]. This characterization of the infinite disorder fixed point is one of the important ingredients of our analysis., Comment: 49 pages, 4 figures, some changes and corrections with respect to v2

The effect of triangularity on tokamak boundary plasma turbulence is investigated by using global, flux-driven, three-dimensional, two-fluid simulations. The simulations show that negative triangularity stabilizes boundary plasma turbulence, and linear investigations reveal that this is due to a reduction of the magnetic curvature drive of interchange instabilities, such as the resistive ballooning mode. As a consequence, the pressure decay length $L_p$, related to the SOL power fall-off length $\lambda_q$, is found to be affected by triangularity. Leveraging considerations on the effect of triangularity on the linear growth rate and nonlinear evolution of the resistive ballooning mode, the analytical theory-based scaling law for $L_p$ in L-mode plasmas, derived by Giacomin \textit{et al.} [{Nucl. Fusion}, \href{https://doi.org/10.1088/1741-4326/abf8f6}{\textbf{61} 076002} (2021)], is extended to include the effect of triangularity. The scaling is in agreement with nonlinear simulations and a multi-machine experimental database, which include recent TCV discharges dedicated to the study of the effect of triangularity in L-mode diverted discharges. Overall, the present results highlight that negative triangularity narrows the $L_p$ and considering the effect of triangularity is important for a reliable extrapolation of $\lambda_q$ from present experiments to larger devices., Comment: 28 pages, 12 figures, submitted to Plasma Physics and Controlled Fusion

First nonlinear gyrokinetic simulations of microtearing modes in the core of a MAST case are performed on two surfaces of the high-collisionality discharge used in Valovi\v{c} et al. Nucl. Fusion 51.7 (2011) to obtain the favorable energy confinement scaling with collisionality, $\tau_E\propto\,\nu_*^{-1}$. On the considered surfaces microtearing modes dominate linearly at binormal length scales of the order of the ion Larmor radius. While the effect of electron collision frequency is moderate in linear simulations, a strong dependence on this parameter is found in nonlinear simulations at $r/a=0.5$, where $r$ and $a$ are the surface and tokamak minor radius, respectively. The dynamics of magnetic islands generated by microtearing modes is analysed, showing that the radial extent of the stochastic region caused by islands overlapping plays an important role in determining the saturation level of the microtearing mode driven heat flux. Local nonlinear gyrokinetic simulations show that the microtearing mode driven heat flux, $Q_e^\mathrm{MTM}$, is largely dominated by magnetic flutter and depends strongly on the magnetic shear, $\hat{s}$. Comparing two surfaces, $r/a=0.5$ and $r/a=0.6$, reveals that $Q_e^\mathrm{MTM}$ is negligible at $r/a=0.5$ ($\hat{s}=0.34$), with the electron temperature gradient driven heat flux, $Q_e^\mathrm{ETG}$, comparable to the experimental electron heat flux, $Q_e^\mathrm{exp}$, while $Q_e^\mathrm{MTM}$ is significantly larger and comparable to $Q_e^\mathrm{ETG}$ and $Q_e^\mathrm{exp}$ at $r/a=0.6$ ($\hat{s}=1.1$). Microtearing modes cause more experimentally significant transport in higher $\hat{s}$ regions and may influence (together with electron temperature gradient modes) the observed scaling of energy confinement time with collisionality (Valovi\v{c} et al. Nucl. Fusion 51.7 (2011)).

The sixth assessment of the international panel on climate change (IPCC) states that "cumulative net CO2 emissions over the last decade (2010-2019) are about the same size as the 11 remaining carbon budget likely to limit warming to 1.5C (medium confidence)." Such reports directly feed the public discourse, but nuances such as the degree of belief and of confidence are often lost. In this paper, we propose a formal account for allowing such degrees of belief and the associated confidence to be used to label arguments in abstract argumentation settings. Differently from other proposals in probabilistic argumentation, we focus on the task of probabilistic inference over a chosen query building upon Sato's distribution semantics which has been already shown to encompass a variety of cases including the semantics of Bayesian networks. Borrowing from the vast literature on such semantics, we examine how such tasks can be dealt with in practice when considering uncertain probabilities, and discuss the connections with existing proposals for probabilistic argumentation.

We consider a forager diffusing via a fractional heat equation and we introduce several efficiency functionals whose optimality is discussed in relation to the L\'evy exponent of the evolution equation. Several biological scenarios, such as a target close to the forager, a sparse environment, a target located away from the forager and two targets are specifically taken into account. The optimal strategies of each of these configurations are here analyzed explicitly also with the aid of some special functions of classical flavor and the results are confronted with the existing paradigms of the L\'evy foraging hypothesis. Interestingly, one discovers bifurcation phenomena in which a sudden switch occurs between an optimal (but somehow unreliable) L\'evy foraging pattern of inverse square law type and a less ideal (but somehow more secure) classical Brownian motion strategy. Additionally, optimal foraging strategies can be detected in the vicinity of the Brownian one even in cases in which the Brownian one is pessimizing an efficiency functional., Comment: Journal of Mathematical Biology

Two-fluid, three-dimensional, flux-driven, global, electromagnetic turbulence simulations carried out by using the GBS code are used to identify the main parameters controlling turbulent transport in the tokamak boundary and to delineate an electromagnetic phase space of edge turbulence. Four turbulent transport regimes are identified: (i) a regime of fully developed turbulence appearing at intermediate values of collisionality and $\beta$, with turbulence driven by resistive ballooning modes, related to the L-mode operation of tokamaks, (ii) a regime of reduced turbulent transport at low collisionality and large heat source, with turbulence driven by drift-waves, related to a high-density H-mode regime, (iii) a regime of extremely large turbulent transport at high collisionality, which is associated with the crossing of the density limit, and (iv) a regime above the ideal ballooning limit at high $\beta$, with global modes affecting the dynamics of the entire confined region, which can be associated with the crossing of the $\beta$ limit. The transition from the reduced to the developed turbulent transport regime is associated here with the H-mode density limit and an analytical scaling law for maximum edge density achievable in H-mode is obtained. Analogously, analytical scaling laws for the crossing of the L-mode density and $\beta$ limits are provided and compared to the results of GBS simulations.

A first-principles scaling law, based on turbulent transport considerations, and a multi-machine database of density limit discharges from the ASDEX Upgrade, JET and TCV tokamaks, show that the increase of the boundary turbulent transport with the plasma collisionality sets the maximum density achievable in tokamaks. This scaling law shows a strong dependence on the heating power, therefore predicting for ITER a significantly larger safety margin than the Greenwald empirical scaling (Greenwald et al, Nucl. Fusion, 28(12), 1988) in case of unintentional H-L transition.

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball., Comment: Mathematics in Engineering, in press

Results of a three-dimensional, flux-driven, electrostatic, global, two-fluid turbulence simulation for a 5-field period stellarator with an island divertor are presented. The numerical simulation is carried out with the GBS code, recently extended to simulate plasma turbulence in non-axisymmetric magnetic equilibria. The vacuum magnetic field used in the simulation is generated with the theory of Dommaschk potentials, and describes a configuration with a central region of nested flux surfaces, surrounded by a chain of magnetic islands, similarly to the diverted configurations of W7-X. The heat outflowing from the core reaches the island region and is transported along the magnetic islands, striking the vessel walls, which correspond to the boundary of the simulation domain. The radial transport of particles and heat is found to be mainly driven by a field-aligned coherent mode with poloidal number $m=4$. The analysis of this mode, based on non-local linear theory considerations, shows its ballooning nature. In contrast to tokamak simulations and experiments, where blobs often contribute to transport, we do not observe the presence of intermittent transport events.

A new version of GBS (Ricci et al. Plasma Phys. Control. Fusion 54, 124047, 2012; Halpern et al. J. Comput. Phys. 315, 388-408, 2016; Paruta et al. Phys. Plasmas 25, 112301, 2018) is described. GBS is a three-dimensional, flux-driven, global, two-fluid turbulence code developed for the self-consistent simulation of plasma turbulence and kinetic neutral dynamics in the tokamak boundary. In the new version presented here, the simulation domain is extended to encompass the whole plasma volume, avoiding an artificial boundary with the core, hence retaining the core-edge-SOL interplay. A new toroidal coordinate system is introduced to increase the code flexibility, allowing for the simulation of arbitrary magnetic configurations (e.g. single-null, double-null and snowflake configurations), which can also be the result of the equilibrium reconstruction of an experimental discharge. The implementation of a new iterative solver for the Poisson and Amp\`ere equations is presented, leading to a remarkable speed-up of the code with respect to the use of direct solvers, therefore allowing for efficient electromagnetic simulations that avoid the use of the Boussinesq approximation. The self-consistent kinetic neutral model, initially developed for limited configurations, is ported to the magnetic configurations considered by the present version of GBS and carefully optimized. A new MPI parallelisation is implemented to evolve the plasma and neutral models in parallel, thus improving the code scalability. The numerical implementation of the plasma and neutral models is verified by means of the method of manufactured solutions. As an example of the simulation capabilities of the new version of GBS, a simulation of a TCV tokamak discharge is presented.

Detection of low-reflectivity objects can be improved by the so-called quantum illumination procedure. However, quantum detection probability exponentially decays with the source bandwidth. The Josephson Parametric Amplifiers (JPAs) technology utilized as a source, generating a pair of entangled signals called two-mode squeezed vacuum states, shows a very narrow bandwidth limiting the operation of the microwave quantum radar (MQR). In this paper, for the first time, a microwave quantum radar setup based on quantum illumination protocol and using a Josephson Traveling Wave Parametric Amplifier (JTWPA) is proposed. Measurement results of the developed JTWPA, pumped at 12 GHz, show an ultrawide bandwidth equal to 10 GHz at X-band making our MQR a promising candidate for the detection of stealth objects.

Self-consistent full-size turbulent-transport simulations of the divertor and SOL of existing tokamaks have recently become feasible. This enables the direct comparison of turbulence simulations against experimental measurements. In this work, we perform a series of diverted Ohmic L-mode discharges on the TCV tokamak, building a first-of-a-kind dataset for the validation of edge turbulence models. This dataset, referred to as TCV-X21, contains measurements from 5 diagnostic systems -- giving a total of 45 1- and 2-D comparison observables in two toroidal magnetic field directions. The dataset is used to validate three flux-driven 3D fluid-turbulence models: GBS, GRILLIX and TOKAM3X. With each model, we perform simulations of the TCV-X21 scenario, tuning the particle and power source rates to achieve a reasonable match of the upstream separatrix value of density and electron temperature. We find that the simulations match the experimental profiles for most observables at the OMP -- both in terms of profile shape and absolute magnitude -- while a poorer agreement is found towards the divertor targets. The match between simulation and experiment is seen to be sensitive to the value of the resistivity, the heat conductivities, the power injection rate and the choice of sheath boundary conditions. Additionally, despite targeting a sheath-limited regime, the discrepancy between simulations and experiment also suggests that the neutral dynamics should be included. The results of this validation show that turbulence models are able to perform simulations of existing devices and achieve reasonable agreement with experimental measurements. Where disagreement is found, the validation helps to identify how the models can be improved. By publicly releasing the experimental dataset, this work should help to guide and accelerate the development of predictive turbulence simulations of the edge and SOL.

We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments., Comment: 42 pages, 6 figures. Accepted for publication on "Mathematical Physics, Analysis and Geometry"

We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $\log$-H\"older continuous., Comment: 29 pages, 2 figures. Introduction is restructured, a few misprints corrected. Accepted for publication on Alea

Theory-based scaling laws of the near and far scrape-off layer (SOL) widths are analytically derived for L-mode diverted tokamak discharges by using a two-fluid model. The near SOL pressure and density decay lengths are obtained by leveraging a balance among the power source, perpendicular turbulent transport across the separatrix, and parallel losses at the vessel wall, while the far SOL pressure and density decay lengths are derived by using a model of intermittent transport mediated by filaments. The analytical estimates of the pressure decay length in the near SOL is then compared to the results of three-dimensional, flux-driven, global, two-fluid turbulence simulations of L-mode diverted tokamak plasmas, and validated against experimental measurements taken from an experimental multi-machine database of divertor heat flux profiles, showing in both cases a very good agreement. Analogously, the theoretical scaling law for the pressure decay length in the far SOL is compared to simulation results and to experimental measurements in TCV L-mode discharges, pointing out the need of a large multi-machine database for the far SOL decay lengths., Comment: Nucl. Fusion 2021

The results of flux-driven, two-fluid simulations in single-null configurations are used to investigate the processes determining the turbulent transport in the tokamak edge. Three turbulent transport regimes are identified: (i) a developed transport regime with turbulence driven by an interchange instability, which shares a number of features with the standard L-mode of tokamak operation, (ii) a suppressed transport regime, characterized by a higher value of the energy confinement time, low-amplitude relative fluctuations driven by a Kelvin-Helmholtz instability, a strong E x B sheared flow, and the formation of a transport barrier, which recalls the H-mode, and (iii) a degraded confinement regime, characterized by a catastrophically large interchange-driven turbulent transport, which reminds the crossing of the Greenwald density limit.We derive an analytical expression of the pressure gradient length in the three regimes. The transition from the developed to the suppressed transport regime is obtained by increasing the heat source or decreasing the collisionality and vice versa for the transition from the developed transport regime to the degraded confinement regime. An analytical expression of the power threshold to access the suppressed transport regime, linked to the power threshold for H-mode access, as well as the maximum density achievable before entering the degraded confinement regime, related to the Greenwald density, are also derived. The experimental dependencies of the power threshold for H-mode access on density, tokamak major radius, and isotope mass are retrieved. The analytical estimate of the density limit contains the correct dependence on the plasma current and on the tokamak minor radius.

One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials. For some more complex models, notably pinning models based on higher dimensional renewals, it has been shown that other phases may be present. We study a generalization of the one dimensional pinning model in which the energy may depend in a nonlinear way on the contact fraction: this class of models contains the circular DNA case considered in the bio-physics literature. We give a full solution of this generalized pinning model in absence of disorder and show that another transition appears. In fact the systems may display up to three different regimes: delocalization, partial localization and full localization. What happens in the partially localized regime can be explained in terms of the "big-jump" phenomenon for sums of heavy tail random variables under conditioning. We then show that disorder completely smears this second transition and we are back to the delocalization versus localization scenario. In fact we show that the disorder, even if arbitrarily weak, is incompatible with the presence of a big-jump., Comment: 33 pages, 5 figures

We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the localization transition of a lattice free field $\phi=(\phi(x))_{x \in Z^d}$, $d\ge 3$, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian $$ \sum_{x\in Z^d }(\beta \omega_x+h)\delta_x,$$ where $\delta_x=1_{[-1,1]}(\phi(x))$, and $(\omega_x)_{x\in Z^d}$ is an IID centered field. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c(\beta)$: from a delocalized phase $hh_c(\beta)$ where the field sticks to the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of $h$ is identified and it coincides, up to the sign, with the $\log$-Laplace transform of $\omega=\omega_x$, that is $-h_c(\beta)=\lambda(\beta):=\log E[e^{\beta\omega}]$. Here we obtain the sharp critical behavior of the free energy approaching criticality: $$\lim_{u\searrow 0} \frac{ F(\beta,h_c(\beta)+u)}{u^2}= \frac{1}{2\, \textrm{Var}\left(e^{\beta \omega-\lambda(\beta)}\right)}.$$ Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is $\sqrt{2\sigma_d^2\vert\log(h-h_c(\beta))\vert}$ to leading order when $h\searrow h_c(\beta)$ except on a vanishing fraction of sites ($\sigma_d^2$ is the single site variance of the free field)., Comment: 62 pages

The theory of abstract argumentation frameworks (afs) has, in the main, focused on finite structures, though there are many significant contexts where argumentation can be regarded as a process involving infinite objects. To address this limitation, in this paper we propose a novel approach for describing infinite afs using tools from formal language theory. In particular, the possibly infinite set of arguments is specified through the language recognized by a deterministic finite automaton while a suitable formalism, called attack expression, is introduced to describe the relation of attack between arguments. The proposed approach is shown to satisfy some desirable properties which can not be achieved through other "naive" uses of formal languages. In particular, the approach is shown to be expressive enough to capture (besides any arbitrary finite structure) a large variety of infinite afs including two major examples from previous literature and two sample cases from the domains of multi-agent negotiation and ambient intelligence. On the computational side, we show that several decision and construction problems which are known to be polynomial time solvable in finite afs are decidable in the context of the proposed formalism and we provide the relevant algorithms. Moreover we obtain additional results concerning the case of finitary afs.

The issue of representing attacks to attacks in argumentation is receiving an increasing attention as a useful conceptual modelling tool in several contexts. In this paper we present AFRA, a formalism encompassing unlimited recursive attacks within argumentation frameworks. AFRA satisfies the basic requirements of definition simplicity and rigorous compatibility with Dung's theory of argumentation. This paper provides a complete development of the AFRA formalism complemented by illustrative examples and a detailed comparison with other recursive attack formalizations.

We investigate the generalized Poland-Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in Giacomin, Khatib (Stoch. Proc. Appl., 2017), via a $2$-dimensional renewal approach, with a loop exponent $2+\alpha$ (${\alpha>0}$): it was found to undergo a localization/delocalization phase transition of order $\nu = \min(1,\alpha)^{-1}$, together with -- in general -- other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris' predictions for $d$-dimensional disordered systems (here $d=2$). First, we prove that when $\alpha<1$ (i.e. $\nu>d/2$), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order $\nu=\alpha^{-1}$. On the other hand, when $\alpha>1$, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant., Comment: 36 pages, 3 figures

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erd\H{o}s-R\'enyi graph with parameter $p_n\in (0, 1]$, where $n$ is the size of the graph (i.e., the number of particles). If $p_n\equiv 1$ the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as $n\to \infty$ to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erd\H{o}s-R\'enyi graphs with $\lim_n p_n n =\infty$, and properly rescaling the interaction to account for the dilution introduced by $p_n$. However, these results have been proven under strong assumptions on that initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results -- Law of Large Numbers and Large Deviation Principle -- assuming only the convergence of the empirical measure of the initial condition., Comment: 16 pages

We consider a particular weak disorder limit ("continuum limit") of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst [J. Phys. A (1983)], which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy-Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu [Phys. Rev. 1968] and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is $C^\infty$ but not analytic at the critical temperature., Comment: 46 pages, one figure. Introduction reorganized, Proposition 1.5 corrects Proposition 1.6 of v2. Several other scattered modifications

The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $1+ \alpha\geq 1$. It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean $h$ of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of \emph{strong disorder renormalization}. We focus on the case $\alpha=0$ -- the critical value $h_c(\beta)$ of the parameter $h$ is exactly known (for every strength $\beta$ of the disorder) in this case -- and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $h\searrow h_c(\beta)$ the free energy vanishes faster than any power of $h-h_c(\beta)$. But we show that the free energy vanishes much faster than the physicists' prediction., Comment: 26 pages, 1 figure

This work presents an algorithm for changing from latitudinal to longitudinal formation of autonomous aircraft squadrons. The maneuvers are defined dynamically by using a predefined set of 3D basic maneuvers. This formation changing is necessary when the squadron has to perform tasks which demand both formations, such as lift off, georeferencing, obstacle avoidance and landing. Simulations show that the formation changing is made without collision. The time complexity analysis of the transformation algorithm reveals that its efficiency is optimal, and the proof of correction ensures its longitudinal formation features., Comment: Published in: XI Simp\'osio Brasileiro de Automa\c{c}\~ao Inteligente, October, 2013. Fortaleza-CE, Brazil

The Poland-Scheraga model for DNA denaturation, besides playing a central role in applications, has been widely studied in the physical and mathematical literature over the past decades. More recently a natural generalization has been introduced in the biophysics literature to overcome the limits of the original model, namely to allow an excess of bases -- i.e. a different length of the two single stranded DNA chains -- and to allow slippages in the chain pairing. The increased complexity of the model is reflected in the appearance of configurational transitions when the DNA is in double stranded form. In a previous work of two of the authors the generalized Poland-Scheraga model has been analyzed thanks to a representation in terms of a bivariate renewal process. In this work we exploit this representation farther and fully characterize the path properties of the system, making therefore explicit the geometric structures -- and the configurational transitions -- that are observed when the polymer is in the double stranded form. What we prove is that, when the excess of bases is not absorbed in a homogeneous fashion along the double stranded chain, then it either condensates in a single macroscopic loop or it accumulates into an unbound single strand free end., Comment: 29 pages, 5 figures

Optimization - minimization or maximization - in the lattice of subsets is a frequent operation in Artificial Intelligence tasks. Examples are subset-minimal model-based diagnosis, nonmonotonic reasoning by means of circumscription, or preferred extensions in abstract argumentation. Finding the optimum among many admissible solutions is often harder than finding admissible solutions with respect to both computational complexity and methodology. This paper addresses the former issue by means of an effective method for finding subset-optimal solutions. It is based on the relationship between cardinality-optimal and subset-optimal solutions, and the fact that many logic-based declarative programming systems provide constructs for finding cardinality-optimal solutions, for example maximum satisfiability (MaxSAT) or weak constraints in Answer Set Programming (ASP). Clearly each cardinality-optimal solution is also a subset-optimal one, and if the language also allows for the addition of particular restricting constructs (both MaxSAT and ASP do) then all subset-optimal solutions can be found by an iterative computation of cardinality-optimal solutions. As a showcase, the computation of preferred extensions of abstract argumentation frameworks using the proposed method is studied., Comment: Informal proceedings of the 1st Workshop on Trends and Applications of Answer Set Programming (TAASP 2016), Klagenfurt, Austria, 26 September 2016

We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erd\H{o}s-R\'enyi graphs with edge probability $p_n$, $n$ is the number of vertices, such that $\lim_{n \to \infty}p_n n= \infty$. The purpose of this note it twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the $n=\infty$ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with $N$ large but finite, for example the values of $N$ that can be reached in simulations or that correspond to the typical number of interacting units in a biological system., Comment: 14 pages, 1 figure. A few corrections, streamlined some explanations and improved Corollary 1.2

We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the {\guillemotleft}pinning potential{\guillemotright} , that rewards visits to the proximity of the wall. The wall can be soft, i.e. the field has a finite penalty to enter the lower half plane, or hard when the penalty is infinite. In general the pinning potential is disordered and it gives on average a reward h in $\mathbb{R}$ (a negative reward is a penalty): the energetic contribution when the field at site x visits the pinning region is $\beta \omega_x+h$, $\{\omega_x\}_{x \in \mathbb{Z}^d}$ are IID centered and exponentially integrable random variables of unit variance and $\beta\ge 0$. In [E. Bolthausen, J.-D. Deuschel and O. Zeitouni, J. Math. Phys. 41 (2000), 1211-1223] it is shown that, when $\beta=0$ (that is, in the non disordered model), a delocalization-localization transition happens at $h=0$, in particular the free energy of the system is zero for $h \le 0$ and positive for $h>0$. We show that, for $\beta\neq 0$, the transition happens at $h=h_c(\beta):=- \log \mathbb{E} \exp(\beta \omega_x)$ and we find the precise asymptotic behavior of the logarithm of the free energy density of the system when $h \searrow h_c(\beta)$. In particular, we show that the transition is of infinite order in the sense that the free energy is smaller than any power of $h-h_c(\beta)$ in the neighborhood of the critical point and that disorder does not modify at all the nature of the transition. We also provide results on the behavior of the paths of the random field in the limit $N \to \infty$., Comment: 23 pages

We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with distribution $\mu$. For a large class of $\mu$, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16:2641, 1983) that the Lyapunov exponent behaves like $C \varepsilon^{2 \alpha}$ in the limit $\varepsilon \searrow 0$, where $\alpha \in (0,1)$ and $C>0$ are determined by $\mu$. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small $\varepsilon$. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent., Comment: 35 pages, 1 figure, LaTeX. Various revisions including many changes to the introduction, a simpler statement of the results of Section 4, and some simplifications of the calculations in Section 5

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise - that is, we modulate the noise by a factor $\varepsilon \searrow 0$ - and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp\left(c \varepsilon^{-2}\right)$, $c>0$, and we show both that on the time scale $\varepsilon^{-2}$ the "'dephasing" (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated, to leading order, by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle., Comment: 24 pages, 1 figure. Small changes, added four references

The Poland-Scheraga model describes the denaturation transition of two complementary - in particular, equally long - strands of DNA, and it has enjoyed a remarkable success both for quantitative modeling purposes and at a more theoretical level. The solvable character of the homogeneous version of the model is one of features to which its success is due. In the bio-physical literature a generalization of the model, allowing different length and non complementarity of the strands, has been considered and the solvable character extends to this substantial generalization. We present a mathematical analysis of the homogeneous generalized Poland-Scheraga model. Our approach is based on the fact that such a model is a homogeneous pinning model based on a bivariate renewal process, much like the basic Poland-Scheraga model is a pinning model based on a univariate, i.e. standard, renewal. We present a complete analysis of the free energy singularities, which include the localization-delocalization critical point and (in general) other critical points that have been only partially captured in the physical literature. We obtain also precise estimates on the path properties of the model., Comment: 45 pages, 7 figures

The ITER tokamak will be operating with 5 microwave diagnostic systems. While they rely on different physics, they share a common need: transmitting low and high power microwave in the range of 12 GHz to 1000 GHz(different bandwidths for different diagnostics) between the plasma and a diagnostic area tens of meters away. The designs proposed for vacuum windows, in-vessel waveguides and antennas are presented together with the development activities needed to finalise this work., Comment: Part of SOFT 2014 conference, San Sebastian, Spain

This paper provides a rigorous study of the localization transition for a Gaussian free field on $\mathbb{Z}^d$ interacting with a quenched disordered substrate that acts on the interface when the interface height is close to zero. The substrate has the tendency to localize or repel the interface at different sites and one can show that a localization-delocalization transition takes place when varying the average pinning potential $h$: the free energy density is zero in the delocalized regime, that is for $h$ smaller than a threshold $h_c$, and it is positive for $h>h_c$. For $d\ge 3$ we compute $h_c$ and we show that the transition happens at the same value as for the annealed model. However we can show that the critical behavior of the quenched model differs from the one of the annealed one. While the phase transition of the annealed model is of first order, we show that the quenched free energy is bounded above by $ (h-h_c)_+^2$ times a positive constant and that, for Gaussian disorder, the quadratic behavior is sharp. Therefore this provides an example in which a {\sl relevant disorder critical exponent} can be made explicit: in theoretical physics disorder is said to be {\sl relevant} when the disorder changes the critical behavior of a system and, while there are cases in which it is known that disorder is relevant, the exact critical behavior is typically unknown. For $d=2$ we are not able to decide whether the quenched and annealed critical points coincide, but we provide an upper bound for the difference between them., Comment: 55 pages, 1 figure. Added a statement on path properties, corrected misprints and reformulated some arguments