Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.     

Low Density Phases in a Uniformly Charged Liquid

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form \({\gamma=(\langle u_i,v_j\rangle)_{i,j=1}^n}\), where the vectors u_i and v_j are in the unit ball of a real Hilbert space. In this work we study the probability of the nonlocal nature of these correlations as a function of \({\alpha=\frac{m}{n}}\), where the previous vectors are sampled according to the Haar measure in the unit sphere of \({\mathbb R^m}\). In particular, we prove the existence of an \({\alpha_0 > 0}\) such that if \({\alpha\leq \alpha_0}\), \({\gamma}\) is nonlocal with probability tending to 1 as \({n\rightarrow \infty}\), while for \({\alpha > 2}\), \({\gamma}\) is local with probability tending to 1 as \({n\rightarrow \infty}\).     

Decay Behaviors of the <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript> Hadronic Molecules

In this proceeding, we present our recent work on decay behaviors of the P_c hadronic molecules, which can help to disentangle the nature of the two P_c pentaquark-like structures. The results turn out that the relative ratio of the decays of P _c ⁺ (4380) to \({\bar D *}{\Lambda _c}\) and J/ψp is very different for P_c being a \({\bar D *}{\Sigma _c}\) or \(\bar D\Sigma _c *\) bound state with \({J^P} = \frac{{{3 - }}}{2}\) And from the total decay width, we find that P_c(4380) being a \(\bar D\Sigma _c *\) molecule state with \({J^P} = \frac{{{3 - }}}{2}\) and P_c(4450) being a \({\bar D *}{\Sigma _c}\) molecule state with \({J^P} = \frac{{{5 + }}}{2}\) is more favorable to the experimental data.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Evaluation of spectral dispersion of optical constants of a-Se films from their normal-incidence transmittance spectra using Swanepoel algebraic envelope approach

Spectral dispersions of index of refraction \({n(\lambda )}\) and extinction coefficient \({\kappa (\lambda )}\) of undoped amorphous selenium (a-Se) films of three thicknesses (d?≈?0.5, 0.75, and 1.0 µm) were evaluated by analyzing experimental room-temperature normal-incidence transmittance-wavelength (\({{T_{{\text{exp}}}}(\lambda )} - \lambda\)) data (λ =?400–1100 µm) of their air-supported {a-Se film/thick glass slide}-stacks using Swanepoel’s transmission envelope theory of uniform films. Above a wavelength \({{\lambda _c}\,\, \approx \,\,640\;{\text{nm}}}\), as-measured \({{T_{{\text{exp}}}}(\lambda )}\,\, - \,\lambda\) spectra display well-resolved maxima and minima, with minor shrinkage in transparent and weak absorption regions (750–1100 nm). Below \({\lambda _{\text{c}}}\), a smeared sharp decline of \({{T_{{\text{exp}}}}(\lambda )}\) with decreasing λ, signifying strong absorption in a-Se films and existence of band-tail localized states. For λ > λ _c, the \({n\,(\lambda )}\, - \,\lambda\) data retrieved from algebraic envelope procedures followed a Sellmeier-like dispersion relation, with the best-fit values of high-frequency dielectric constant \({{\varepsilon _\infty }\, \approx \,\,{\text{4.9}}}\), static index of refraction \({{n_{\text{0}}} = n\left( {E\, \to \,{\text{0}}} \right)\,\, \approx \,\,{\text{2.43}}}\), and resonance wavelength \({{\lambda _0}\, \approx \,490\,\,{\text{nm}}}\), which may be assigned to onset of photogeneration in a-Se. Urbach-like dependency of absorption coefficient \({\alpha (h{{\nu }})}\) of a-Se films on photon energy \({h{{\nu }}}\) was realized with an Urbach-tail breadth of 85 meV. All achieved optical parameters were found to be slightly dependent on film thickness. Findings of present algebraic analysis are consistent with reported literature results obtained on the basis of other optical analytical approaches.     

Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps

There is a general method for constructing a soliton hierarchy from a splitting \({{L_{\pm}}}\) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \({\mathcal{L}_{+}}\). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup \({L_-}\) a solution \({u_{f}}\) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \({\tau_{f}}\) for each element \({f \in L_-}\). In this paper, we give integral formulas for variations of \({\ln\tau_{f}}\) and second partials of \({\ln\tau_{f}}\), discuss whether we can recover solutions \({u_{f}}\) from \({\tau_{f}}\), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \({GL(n,\mathbb{C})}\)-hierarchy.     

Positive Casimir and Central Characters of Split Real Quantum Groups

We consider the one parameter family \({\alpha \mapsto T_{\alpha}}\) (\({\alpha \in [0,1)}\)) of Pomeau-Manneville type interval maps \({T_{\alpha}(x) = x(1+2^{\alpha} x^{\alpha})}\) for \({x \in [0,1/2)}\) and \({T_{\alpha}(x)=2x-1}\) for \({x \in [1/2, 1]}\), with the associated absolutely continuous invariant probability measure \({\mu_{\alpha}}\). For \({\alpha \in (0,1)}\), Sarig and Gouëzel proved that the system mixes only polynomially with rate \({n^{1-1/{\alpha}}}\) (in particular, there is no spectral gap). We show that for any \({\psi \in L^{q}}\), the map \({\alpha \to \int_0^{1} \psi\, d \mu_{\alpha}}\) is differentiable on \({[0,1-1/q)}\), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For \({\alpha \ge 1/2}\) we need the \({n^{-1/{\alpha}}}\) decorrelation obtained by Gouëzel under additional conditions.     

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Let \({T=\mathbb R^d}\) . Let a function \({QT^2\to\mathbb C}\) satisfy \({Q(s,t)=\overline{Q(t,s)}}\) and \({|Q(s,t)|=1}\). A generalized statistics is described by creation operators \({\partial_t^\dagger}\) and annihilation operators ? _t , \({t\in T}\), which satisfy the Q-commutation relations: \({\partial_s\partial^\dagger_t = Q(s, t)\partial^\dagger_t\partial_s+\delta(s, t)}\) , \({\partial_s\partial_t = Q(t, s)\partial_t\partial_s}\), \({\partial^\dagger_s\partial^\dagger_t = Q(t, s)\partial^\dagger_t\partial^\dagger_s}\). From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which Q(s, t) is equal to q if s < t, and to \({\bar q}\) if s > t. Here \({q\in\mathbb C}\) , |q| = 1. We start the paper with a detailed discussion of a Q-Fock space and operators \({(\partial_t^\dagger,\partial_t)_{t\in T}}\) in it, which satisfy the Q-commutation relations. Next, we consider a noncommutative stochastic process (white noise) \({\omega(t)=\partial_t^\dagger+\partial_t+\lambda\partial_t^\dagger\partial_t}\) , \({t\in T}\) . Here \({\lambda\in\mathbb R}\) is a fixed parameter. The case λ = 0 corresponds to a Q-analog of Brownian motion, while λ ≠ 0 corresponds to a (centered) Q-Poisson process. We study Q-Hermite (Q-Charlier respectively) polynomials of infinitely many noncommutatative variables \({(\omega(t))_{t\in T}}\) . The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define Q-cumulants of a field \({(\xi(t))_{t\in T}}\). This allows us to define a Q-Lévy process as a field \({(\xi(t))_{t\in T}}\) whose values at different points of T are Q-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a Q-Lévy process, and derive a Nualart–Schoutens-type chaotic decomposition for such a process.     

On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I

We study the determinant \({\det(I-\gamma K_s), 0 < \gamma < 1}\) , of the integrable Fredholm operator K _s acting on the interval (?1, 1) with kernel \({K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}\) . This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature \({\beta=2}\) , in the presence of an external potential \({v=-\frac{1}{2}\ln(1-\gamma)}\) supported on an interval of length \({\frac{2s}{\pi}}\) . We evaluate, in particular, the double scaling limit of \({\det(I-\gamma K_s)}\) as \({s\rightarrow\infty}\) and \({\gamma\uparrow 1}\) , in the region \({0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}\) , for any fixed \({0 < \delta < 1}\) . This problem was first considered by Dyson (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131–146, 1995).     

$${\mathbb{Z}_N}$$ -Curves Possessing No Thomae Formulae of Bershadsky–Radul Type

A \({\mathbb{Z}_N}\) -curve is one of the form \({y^{N}=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{s})^{m_{s}}}\) . When N = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ₁, . . . , λ _s . In his work on Fermionic fields on \({\mathbb{Z}_N}\) -curves with arbitrary N, Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m _i equal 1. This work was continued by several authors and new Thomae’s type formulae for \({\mathbb{Z}_N}\) -curves with other rotation numbers m _i were found. In this article we prove that for some choices of the rotation numbers the corresponding \({\mathbb{Z}_N}\) -curves do not admit such generalized Thomae’s formulae.     

Off-Diagonal Decay of Toric Bergman Kernels

We study the off-diagonal decay of Bergman kernels \({\Pi_{h^k}(z,w)}\) and Berezin kernels \({P_{h^k}(z,w)}\) for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) where \({D(z,w)}\) is the diastasis. When the metric is only \({C^{\infty}}\) this asymptotic cannot hold for all \({(z,w)}\) since the diastasis is not even defined for all \({(z,w)}\) close to the diagonal. Our main result is that for general toric \({C^{\infty}}\) metrics, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) as long as w lies on the \({\mathbb{R}_+^m}\)-orbit of z, and for general \({(z,w)}\), \({{\rm lim\,sup}_{k \to \infty} \frac{1}{k} {\rm log} P_{h^k}(z,w) \,\leq\, - D(z^*,w^*)}\) where \({D(z, w^*)}\) is the diastasis between z and the translate of w by \({(S^1)^m}\) to the \({\mathbb{R}_+^m}\) orbit of z. These results are complementary to Mike Christ’s negative results showing that \({P_{h^k}(z,w)}\) does not have off-diagonal exponential decay at “speed” k if \({(z,w)}\) lies on the same \({(S^1)^m}\)-orbit.     

Constraining the electric charges of some astronomical bodies in Reissner–Nordström spacetimes and generic <Emphasis Type="Italic">r</Emphasis><Superscript>−2</Superscript>-type power-law potentials from orbital motions

We put independent model dynamical constraints on the net electric charge Q of some astronomical and astrophysical objects by assuming that their exterior spacetimes are described by the Reissner-Nordström, metric, which induces an additional potential \({U_{\rm RN} \propto Q^2 r^{-2}}\). From the current bounds \({\Delta \dot \varpi}\) on any anomalies in the secular perihelion rate \({\dot \varpi}\) of Mercury and the Earth–mercury ranging Δρ, we have \({\left|Q_{\odot}\right| \lesssim 1-0.4 \times 10^{18}\ {\rm C}}\). Such constraints are ~60–200 times tighter than those recently inferred in literature. For the Earth, the perigee precession of the Moon, determined with the Lunar Laser Ranging technique, and the intersatellite ranging Δρ for the GRACE mission yield \({\left|Q_{\oplus} \right| \lesssim 5-0.4 \times 10^{14}\ {\rm C}}\). The periastron rate of the double pulsar PSR J0737-3039A/B system allows to infer \({\left|Q_{\rm NS} \right| \lesssim 5\times 10^{19}\ {\rm C}}\). According to the perinigricon precession of the main sequence S2 star in Sgr A^*, the electric charge carried by the compact object hosted in the Galactic Center may be as large as \({\left|Q_{\bullet} \right| \lesssim 4\times 10^{27} \ {\rm C}}\). Our results extend to other hypothetical power-law interactions inducing extra-potentials \({U_{\rm pert} = \Psi r^{-2}}\) as well. It turns out that the terrestrial GRACE mission yields the tightest constraint on the parameter \({\Psi}\), assumed as a universal constant, amounting to \({|\Psi| \lesssim 5\times 10^{9}\ {\rm m^4\ s^{-2}}}\).     

Unique Continuation from Infinity in Asymptotically Anti-de Sitter Spacetimes

We consider the unique continuation properties of asymptotically anti-de Sitter spacetimes by studying Klein–Gordon-type equations \({\Box_g \phi + \sigma \phi = {\mathcal{G}} ( \phi, \partial \phi )}\), \({\sigma \in {\mathbb{R}}}\), on a large class of such spacetimes. Our main result establishes that if \({\phi}\) vanishes to sufficiently high order (depending on \({\sigma}\)) on a sufficiently long time interval along the conformal boundary \({{\mathcal{I}}}\), then the solution necessarily vanishes in a neighborhood of \({{\mathcal{I}}}\). In particular, in the \({\sigma}\)-range where Dirichlet and Neumann conditions are possible on \({{\mathcal{I}}}\) for the forward problem, we prove uniqueness if both these conditions are imposed. The length of the time interval can be related to the refocusing time of null geodesics on these backgrounds and is expected to be sharp. Some global applications as well as a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.     

Crossover to the Stochastic Burgers Equation for the WASEP with a Slow Bond

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter \({\rho \in (0,1)}\). The rate of passage of particles to the right (resp. left) is \({\frac{1}{2} + \frac{a}{2n^{\gamma}}}\) (resp. \({\frac{1}{2} - \frac{a}{2n^{\gamma}}}\)) except at the bond of vertices \({\{-1,0\}}\) where the rate to the right (resp. left) is given by \({\frac{\alpha}{2n^\beta} + \frac{a}{2n^{\gamma}}}\) (resp. \({\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\gamma}}}\)). Above, \({\alpha > 0}\), \({\gamma \geq \beta \geq 0}\), \({a\geq 0}\). For \({\beta < 1}\), we show that the limit density fluctuation field is an Ornstein–Uhlenbeck process defined on the Schwartz space if \({\gamma > \frac{1}{2}}\), while for \({\gamma = \frac{1}{2}}\) it is an energy solution of the stochastic Burgers equation. For \({\gamma \geq \beta =1}\), it is an Ornstein–Uhlenbeck process associated to the heat equation with Robin’s boundary conditions. For \({\gamma \geq \beta > 1}\), the limit density fluctuation field is an Ornstein–Uhlenbeck process associated to the heat equation with Neumann’s boundary conditions.     

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H _xy , indexed by \({x,y \in \Lambda \subset \mathbb{Z}^d}\), are independent, uniformly distributed random variables if \({\lvert{x-y}\rvert}\) is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales \({t\ll W^{d/3}}\). We also show that the localization length of the eigenvectors of H is larger than a factor W ^d/6 times the band width. All results are uniform in the size \({\lvert{\Lambda}\rvert}\) of the matrix.     

Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its continuum limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL _N , goes over in the continuum limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL _N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL _N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL _N image represented in the spin-chain. On the continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).     

Meixner Class of Non-Commutative Generalized Stochastic Processes with Freely Independent Values I. A Characterization

Let T be an underlying space with a non-atomic measure σ on it (e.g. \({T=\mathbb R^d}\) and σ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of T, with freely independent values. Such a process (field), ω = ω(t), \({t\in T}\) , is given a rigorous meaning through smearing out with test functions on T, with \({\int_T \sigma(dt)f(t)\omega(t)}\) being a (bounded) linear operator in a full Fock space. We define a set CP of all continuous polynomials of ω, and then define a non-commutative L ²-space L ²(τ) by taking the closure of CP in the norm \({\|P\|_{L^2(\tau)}:=\|P\Omega\|}\) , where Ω is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between L ²(τ) and a (Fock-space-type) Hilbert space \({\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,\gamma_n)}\) , with explicitly given measures γ _n . We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set CP invariant. (Note that, in the general case, the projection of a continuous monomial of order n onto the n ^th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions λ and η ≥ 0 on T, such that, in the \({\mathbb F}\) space, ω has representation \({\omega(t)=\partial_t^\dagger+\lambda(t)\partial_t^\dagger\partial_t+\partial_t+\eta(t)\partial_t^\dagger\partial^2_t}\) , where \({\partial_t^\dagger}\) and ? _t are the usual creation and annihilation operators at point t.     

Graph-Counting Polynomials for Oriented Graphs

If \(\mathcal{F}\) is a set of subgraphs F of a finite graph E we define a graph-counting polynomial \(p_\mathcal{F}(z)=\sum _{F\in \mathcal{F}}z^{|F|}\) In the present note we consider oriented graphs and discuss some cases where \(\mathcal{F}\) consists of unbranched subgraphs E. We find several situations where something can be said about the location of the zeros of \(p_\mathcal{F}\).     

Blow-up of Critical Besov Norms at a Potential Navier–Stokes Singularity

We prove that if an initial datum to the incompressible Navier–Stokes equations in any critical Besov space \({\dot B^{-1+\frac 3p}_{p,q}({\mathbb {R}}^{3})}\), with \({3 < p, q < \infty}\), gives rise to a strong solution with a singularity at a finite time \({T > 0}\), then the norm of the solution in that Besov space becomes unbounded at time T. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) concerning suitable weak solutions blowing up in \({L^{3}({\mathbb R}^{3})}\). Our proof uses profile decompositions and is based on our previous work (Gallagher et al., Math. Ann. 355(4):1527–1559, 2013), which provided an alternative proof of the \({L^{3}({\mathbb R}^{3})}\) result. For very large values of p, an iterative method, which may be of independent interest, enables us to use some techniques from the \({L^{3}({\mathbb R}^{3})}\) setting.     

T-Duality Simplifies Bulk-Boundary Correspondence

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice \({{{\mathbb{Z}}}^{4}}\), for the weakly coupled n-component \({|\varphi|^{4}}\) spin model for all \({n \ge 1}\), and for the continuous-time weakly self-avoiding walk. For the \({|\varphi|^{4}}\) model, we prove that the critical two-point function has |x|^?2 (Gaussian) decay asymptotically, for \({n \ge 1}\). We also determine the asymptotic decay of the critical correlations of the squares of components of \({\varphi}\), including the logarithmic corrections to Gaussian scaling, for \({n \ge 1}\). The above extends previously known results for n = 1 to all \({n \ge 1}\), and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the “watermelon” network consisting of p weakly mutually- and self-avoiding walks, for all \({p \ge 1}\), including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove the existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the \({|\varphi|^{4}}\) model, and provides a unified treatment of both models, and of all the above results.