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Abstract: Abstract We prove convergence of the 2- and 4-point fermionic observables of the FK-Ising model on simply connected domains discretised by a planar isoradial lattice in massive (near-critical) scaling limit. The former is alternatively known as a (fermionic) martingale observable (MO) for the massive interface, and in particular encapsulates boundary visit probabilties of the interface. The latter encodes connection probabilities in the 4-point alternating (generalised Dobrushin) boundary condition, whose exact convergence is then further analysed to yield crossing estimates for general boundary conditions. Notably, we obtain a massive version of the so-called Russo-Seymour-Welsh (RSW) type estimates on isoradial lattice. These observables satisfy a massive version of s-holomorphicity Smirnov (Ann. Math. 172: 1435-1467, 2007), and we develop robust techniques to exploit this condition which do not require any regularity assumption of the domain or a particular direction of perturbation. Since many other near-critical observables satisfy the same relation (cf. Beffara (Ann. Probab. 40: 2667-2689, 2012), Chelkak (arXiv:2104.12858, 2021), Park (Massive Scaling Limit of the Ising Model: Subcritical Analysis and Isomonodromy, 2019)), these strategies are of direct use in the analysis of massive models in broader setting. PubDate: 2022-12-01
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Abstract: Abstract We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime (M, g) with an unknown metric g. We consider measurements done in a neighbourhood \(V\subset M\) of a timelike path \(\mu \) that connects a point \(x^-\) to a point \(x^+\) . The measurements are modelled by a source-to-solution map, which maps a source supported in V to the restriction of the solution to the Boltzmann equation to the set V. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set \(I^+(x^-)\cap I^-(x^+)\subset M\) . The set \(I^+(x^-)\cap I^-(x^+)\) is the intersection of the future of the point \(x^-\) and the past of the point \(x^+\) , and hence is the maximal set to where causal signals sent from \(x^-\) can propagate and return to the point \(x^+\) . The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem. PubDate: 2022-12-01
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Abstract: Abstract We consider the Lorentz gas model of category A (that is, with no corners and of finite horizon) on aperiodic repetitive tilings of \(\mathbb {R}^2\) of finite local complexity. We show that the compact factor of the collision map has the K property. From this we derive properties of the corresponding aperiodic Lorentz gases defined on the plane, such as mixing for pattern equivariant observables and a version of Birkhoff ergodic theorem for the Lorentz gas flow. PubDate: 2022-12-01
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Abstract: Abstract We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin–Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al. (Geom Topol 23:781–864, 2019. https://doi.org/10.2140/gt.2019.23.781. arXiv:1705.06085) if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al. (Commun Math Phys 321:543–575, 2013. https://doi.org/10.1007/s00220-013-1723-0. arXiv:1203.4568) and find agreement. PubDate: 2022-12-01
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Abstract: Abstract We use the periodic Schur process, introduced in (Borodin in Duke Math J 140(3):391–468 2007), to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the \(q^{{{\text {vol}}}}\) measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes. PubDate: 2022-12-01
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Abstract: Abstract Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we show how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associated to a repulsive potential by a series of Darboux transformations. Even though, by its non translation invariance, our model avoids some of the difficulties related to the effect that translation has on virial inequalities of the kink stability problem for wave equations, it still is a classical model and it retains some of the main difficulties. PubDate: 2022-12-01
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Abstract: Abstract We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on \({\mathbb {R}}^m\) are replaced by non-commutative laws of m-tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu’s non-commutative \(L^2\) -Wasserstein distance using a new type of convex functions. As a consequence, we show that if (X, Y) is a pair of optimally coupled m-tuples of non-commutative random variables in a tracial \(\mathrm {W}^*\) -algebra \(\mathcal {A}\) , then \(\mathrm {W}^*((1 - t)X + tY) = \mathrm {W}^*(X,Y)\) for all \(t \in (0,1)\) . Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m-tuples is not separable with respect to the Wasserstein distance for \(m > 1\) . PubDate: 2022-12-01
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Abstract: Abstract The present paper extends the landscape theory pioneered in Filoche and Mayboroda (Proc Natl Acad Sci USA 109(37):14761–14766, 2012), Arnold et al. (Commun Partial Differ Equ 44(11):1186–1216, 2019) and David et al. (Adv Math 390:107946, 2021) to the tight-binding Schrödinger operator on \({\mathbb {Z}}^d\) . In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape. PubDate: 2022-12-01
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Abstract: Abstract One of the main difficulties in proving convergence of discrete models of surface growth to the Kardar–Parisi–Zhang (KPZ) equation in dimensions higher than one is that the correct way to take a scaling limit, so that the limit is nontrivial, is not known in a rigorous sense. To understand KPZ growth without being hindered by this issue, this article introduces a notion of ‘local KPZ behavior’, which roughly means that the instantaneous growth of the surface at a point decomposes into the sum of a Laplacian term, a gradient squared term, a noise term that behaves like white noise, and a remainder term that is negligible compared to the other three terms and their sum. The main result is that for a general class of surfaces, which contains the model of directed polymers in a random environment as a special case, local KPZ behavior occurs under arbitrary scaling limits, in any dimension. PubDate: 2022-12-01
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Abstract: Abstract Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints. PubDate: 2022-12-01
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Abstract: Abstract The statistical model of crystal melting represents BPS configurations of D-branes on a toric Calabi–Yau three-fold. Recently it has been noticed that an infinite-dimensional algebra, the quiver Yangian, acts consistently on the crystal-melting configurations. We physically derive the algebra and its action on the BPS states, starting with the effective supersymmetric quiver quantum mechanics on the D-brane worldvolume. This leads to remarkable combinatorial identities involving equivariant integrations on the moduli space of the quantum mechanics, which can be checked by numerical computations. PubDate: 2022-12-01
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Abstract: Abstract We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair et al. (Commun Math Phys 386(1):253–268, 2021) in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions. PubDate: 2022-12-01
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Abstract: Abstract We consider the Green’s function \(G(z):=(H-z)^{-1}\) of Hermitian random band matrix H on the d-dimensional lattice \((\mathbb {Z}/L\mathbb {Z})^d\) . The entries \(h_{xy}=\overline{h}_{yx}\) of H are independent centered complex Gaussian random variables with variances \(s_{xy}=\mathbb E h_{xy} ^2\) , which satisfy a banded profile so that \(s_{xy}\) is negligible if \( x-y \) exceeds the band width W. For any fixed \(n\in \mathbb {N}\) , we construct an expansion of the T-variable, \(T_{xy}= m ^2 \sum _{\alpha }s_{x\alpha } G_{\alpha y} ^2\) , with an error \({{\,\mathrm{O}\,}}(W^{-nd/2})\) , and use it to prove a local law on the Green’s function. This T-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions \(d\geqslant 8\) in part I (Yang et al. in Delocalization and quantum diffusion of random band matrices in high dimensions I: self-energy renormalization, 2021. arXiv:2104.12048) of this series. PubDate: 2022-12-01
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Abstract: Abstract Motivated by M-theory, we study rank n K-theoretic Donaldson–Thomas theory on a toric threefold X. In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi–Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on \(A_{n-1}\) ALE space, thus extending the usual geometric engineering dictionary to \(n>1\) . We finally speculate about implications for instanton counting on Taub-NUT. PubDate: 2022-12-01
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Abstract: Abstract Motivated by the results from the classical probability theory, we introduce the concepts of tangency and weak domination of noncommutative martingales. Then we establish the weak-type and strong-type estimates arising in this context. The proof rests on a novel Gundy-type decomposition which is of independent interest. We also show the corresponding square function inequalities under the assumption of the weak domination, which extends Pisier and Xu’s Burkholder–Gundy inequalities. The results strengthen and extend the very recent works on noncommutative differentially subordinate martingales, which in turn, give rise to a new application in harmonic analysis: a weak-type estimate (along with a completely bounded version) for the directional Hilbert transform associated with quantum tori. PubDate: 2022-12-01
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Abstract: Abstract This is the first instance in the extensive literature of more than three decades for complete normal form classification of a singular family on a 2n-dimensional center manifold for arbitrary n. We are concerned with complete normal form characterization and classification of non-resonant n-tuple Hopf singular differential systems with radial and rotational nonlinearities. Our analysis is facilitated by using several reduction techniques. These include an invariant cell-decomposition of the state space, a family of smooth flow-invariant foliations, leaf-reduction of differential systems and leaf-normal forms. Each leaf of the foliations is a minimal flow-invariant realization of the state space for all radial and rotational differential systems. Complete simplest normal form characterization for singular flows are provided using a family of leaf-reductions and infinite level (simplest) formal leaf-normal forms. In this direction, we introduce Lie algebra structures on invariant leaf manifolds for the local leaf-normal classifications. Since leaf-manifolds foliate the state space, leaf normal forms for all flow-invariant leaves are required for a complete normal form characterization of the 2n-dimensional system. Thus, we further discuss the geometry and spectral impact of leaf variations on the infinite level leaf-normal form coefficients, leaf-finite determinacy and leaf-universal unfoldings. There are infinitely many leaf-systems for such a 2n-dimensional system. However, we show that a 2n-dimensional system can admit at most a finite number of topologically non-equivalent leaf-normal form systems. These are the ones that classify the 2n-dimensional singular family. PubDate: 2022-12-01
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Abstract: Abstract For the field \(\mathbb {K}= \mathbb {R}\) or \(\mathbb {C}\) , and an integrable distribution \(F \subseteq T_M \otimes _{\mathbb {R}} \mathbb {K}\) on a smooth manifold M, we study the Hochschild cohomology of the dg manifold \((F[1],d_F)\) and establish a canonical isomorphism with the Hochschild cohomology of the algebra of functions on leaf space in terms of transversal polydifferential operators of F. In particular, for the dg manifold \((T_X^{0,1}[1],{\bar{\partial }})\) associated with a complex manifold X, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomology \({H\!H}^{\bullet } (X)\) of the complex manifold X. As an application, we show that the Duflo-Kontsevich type theorem for the dg manifold \((T_X^{0,1}[1],{\bar{\partial }})\) implies the Duflo-Kontsevich theorem for complex manifolds. PubDate: 2022-12-01
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Abstract: Abstract We prove an analog of the Künneth formula for the groups of minimal non-degenerate extensions (Lan et al. in Commun Math Phys 351:709–739, 2017) of symmetric fusion categories. We describe in detail the structure of the group of minimal extensions of a pointed super-Tannakian fusion category. This description resembles that of the third cohomology group of a finite abelian group. We explicitly compute this group in several concrete examples. PubDate: 2022-12-01
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Abstract: Abstract We study complex Dirac structures, that is, Dirac structures in the complexified generalized tangent bundle. These include presymplectic foliations, transverse holomorphic structures, CR-related geometries and generalized complex structures. We introduce two invariants, the order and the (normalized) type. We show that, together with the real index, they allow us to obtain a pointwise classification of complex Dirac structures. For constant order, we prove the existence of an underlying real Dirac structure, which generalizes the Poisson structure associated to a generalized complex structure. For constant real index and order, we prove a splitting theorem, which gives a local description in terms of a presymplectic leaf and a small transversal. PubDate: 2022-12-01
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Abstract: Abstract In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with \(a=b=c=1\) on the square lattice), or, equivalently, of the uniform random homomorphisms from \(\mathbb {Z}^2\) to \(\mathbb {Z}\) , has logarithmic variance. This establishes a strong form of roughness of this height function. PubDate: 2022-12-01
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