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We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski' (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai's trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is two-sided, the observables depend on the full trajectory, both the future and the past. [ABSTRACT FROM AUTHOR]

We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work (Raphaël and Rodnianski in Publ Math Inst Hautes Études Sci 115:1–122, 2012) of Raphaël and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time with the blow-up rate λ (t) = (T - t) e - | log (T - t) | + O (1) . In this paper, we show that this e O (1) -factor in fact converges to the universal constant 2 e - 1 , and hence these solutions contract at the universal rate λ (t) = 2 e - 1 (T - t) e - | log (T - t) | (1 + o t → T (1)) . Our proof is inspired by recent works on type-II blow-up dynamics for parabolic equations. The key improvement is in the construction of an explicit invariant subspace decomposition for the linearized operator perturbed by the scaling generator in the dispersive case, from which we obtain a more precise ODE system determining λ (t) . [ABSTRACT FROM AUTHOR]

In this paper we develop a Nash-Moser iteration type reducibility approach to prove the (inverse) localization for some d-dimensional discrete almost-periodic operators with power-law long-range hopping. We also provide a quantitative lower bound on the regularity of the hopping. As an application, some results of Sarnak (Comm Math Phys 84(3):377–401, 1982), Pöschel (Comm Math Phys 88(4):447–463, 1983), Craig (Comm Math Phys 88(1):113–131, 1983) and Bellissard et al. (Comm Math Phys 88(4):465–477, 1983) are generalized to the power-law hopping case. [ABSTRACT FROM AUTHOR]

The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu (in Commun Math Phys 350(2):699–747, 2017) via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect. [ABSTRACT FROM AUTHOR]

We consider the sine-Gordon (SG) equation in 1 + 1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and ϕ 4 at the level of linear Bäcklund transformations. [ABSTRACT FROM AUTHOR]

The lace expansion for the Ising two-point function was successfully derived in (Sakai in Commun Math Phys 272:283–344, 2007, Proposition 1.1). It is an identity that involves an alternating series of lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster x-space decay (as the two-point function cubed) above the critical dimension d c ( = 4 for finite-variance models) if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of (Sakai in Commun Math Phys 272:283–344, 2007, Lemma 4.2), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic (Sakai 2007, Lemma 4.2), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in (Sakai 2007, Proposition 4.1) but nonetheless obey the same fast decay above the critical dimension d c . Consequently, the lace-expansion results for the Ising and φ 4 models in the literature are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients. [ABSTRACT FROM AUTHOR]

The Weyl law of the Laplacian on the flat torus T n is concerning the number of eigenvalues ≤ λ 2 , which is equivalent to counting the lattice points inside the ball of radius λ in R n . The leading term in the Weyl law is c n λ n , while the sharp error term O (λ n - 2) is only known in dimension n ≥ 5 . Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators H V = - Δ g + V in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531)) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau's heat kernel estimates, Blair–Sire–Sogge's eigenfunction estimates, and Duhamel's principle for the wave equation. [ABSTRACT FROM AUTHOR]

We reconsider the unique continuation property for a general class of tensorial Klein–Gordon equations of the form □ g ϕ + σ ϕ = G (ϕ , ∇ ϕ) , σ ∈ R on a large class of asymptotically anti-de-Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and the second author (Holzegel and Shao in Commun Math Phys 347(3):723–775, 2016; Commun Partial Differ Equ 42(12):1871–1922, 2017; McGill and Shao in Class Quantum Gravity 38(5):054001, 2021) (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways: We replace the so-called null convexity criterion—the key geometric assumption on the conformal boundary needed in McGill and Shao (2021) to establish the unique continuation properties—by a more general criterion that is also gauge invariant. Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary. Similar to McGill and Shao (2021), we connect the failure of our generalized criterion to the existence of certain null geodesics near the conformal boundary. These geodesics are closely related to the classical Alinhac-Baouendi counterexamples to unique continuation (Alinhac and Baouendi in Math Z 220(4):561–568, 1995). Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and the second author (Holzegel and Shao in Unique continuation for the Einstein equations in asymptotically anti-de sitter spacetimes (in preparation), 2022). [ABSTRACT FROM AUTHOR]

In this paper we study the large N limit of the O(N)-invariant linear sigma model, which is a vector-valued generalization of the Φ 4 quantum field theory, on the three dimensional torus. We study the problem via its stochastic quantization, which yields a coupled system of N interacting SPDEs. We prove tightness of the invariant measures in the large N limit. For large enough mass or small enough coupling constant, they converge to the (massive) Gaussian free field at a rate of order 1 / N with respect to the Wasserstein distance. We also obtain tightness results for certain O(N) invariant observables. These generalize some of the results in Shen et al. (Ann Probab 50(1):131–202, 2022) from two dimensions to three dimensions. The proof leverages the method recently developed by Gubinelli and Hofmanová (Commun Math Phys 384(1):1–75, 2021) and combines many new techniques such as uniform in N estimates on perturbative objects as well as the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we establish local well-posedness for the Zakharov system on T d , d ≥ 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to ε -loss of regularity when d = 3 and d ≥ 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter's range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies supp f ^ ⊂ { (ξ , τ) ∈ R d + 1 : 1 - 1 N ≤ | ξ | ≤ 1 + 1 N , | τ - | ξ | 2 | ≤ 1 N 2 } for large N ≥ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we establish the full L p boundedness of noncommutative Bochner–Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in Chen et al. (Commun Math Phys 322(3):755–805, 2013) in the sense of the L p convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner–Riesz means. [ABSTRACT FROM AUTHOR]

In this paper we study the regularity of the non-cutoff Vlasov–Poisson–Boltzmann system for plasma particles of two species in the whole space R 3 with hard potential. The existence of global-in-time nearby Maxwellian solutions is known for soft potential from Duan and Liu (Commun Math Phys 324(1):1–45, 2013). However the smoothing effect of these solutions has been a challenging open problem. We establish the global existence and regularizing effect to the Cauchy problem for hard potential with large time decay. Hence, the solutions are smooth with respect to (t, x, v) for any positive time t > 0 . This gives regularity to the Vlasov–Poisson–Boltzmann system, which enjoys a similar smoothing effect as the Boltzmann equation. The proof is based on the time-weighted energy method building also upon the pseudo-differential calculus. [ABSTRACT FROM AUTHOR]

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of Kirillov and Ostrik (Adv Math 171(2):183–227, 2002) in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by Davydov (J Algebra 323(5):1321–1348, 2010) to arbitrary characteristic. [ABSTRACT FROM AUTHOR]

We exhibit non-unique Leray solutions of the forced Navier–Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in Albritton et al. (Ann Math 196(1):415–455, 2022), the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as t → 0 + . In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik (Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, 2018; Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018) to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in Albritton et al. (2022). [ABSTRACT FROM AUTHOR]

In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics. [ABSTRACT FROM AUTHOR]

We prove that poly (t) · n 1 / D -depth local random quantum circuits with two qudit nearest-neighbor gates on a D-dimensional lattice with n qudits are approximate t-designs in various measures. These include the "monomial" measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was poly (t) · n due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for D = 1 . We also improve the "scrambling" and "decoupling" bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ( PH ) is infinite and that certain counting problems are # P -hard "on average", sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called "anti-concentration", meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that O (n) depth suffices for anti-concentration. The proof is based on a previous construction of t-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size O (n ln 2 n) corresponding to depth O (ln 3 n) . We also show a lower bound of Ω (n ln n) for the size of such circuit in this case. We also prove that anti-concentration is possible in depth O (ln n ln ln n) (size O (n ln n ln ln n) ) using a different model. [ABSTRACT FROM AUTHOR]

We derive residue formulas for the regularized integrals (introduced by Li and Zhou in Commun Math Phys 388:1403–1474, 2021) on configuration spaces of elliptic curves. Based on these formulas, we prove that the regularized integrals satisfy holomorphic anomaly equations, providing a mathematical formulation of the so-called contact term singularities. We also discuss residue formulas for the ordered A-cycle integrals and establish their relations with those for the regularized integrals. [ABSTRACT FROM AUTHOR]

We prove here that the height function associated to non-simple exclusion processes with arbitrary jump-length converges to the solution of the Kardar–Parisi–Zhang SPDE under suitable scaling and renormalization. This extends the work of Dembo and Tsai (Commun Math Phys 341(1):219–261, 2016) for arbitrary jump-length and Goncalves and Jara (Stoch Process Appl 127(12):4029–4052, 2017) for the non-stationary regime. Thus we answer a "Big Picture Question" from the AIM workshop on KPZ and also expand on the almost empty set of non-integrable and non-stationary particle systems for which weak KPZ universality is proven. We use an approximate microscopic Cole-Hopf transform like in Dembo and Tsai (2016) but we develop tools to analyze local statistics of the particle system via local equilibrium and work of Goncalves and Jara (2017). Local equilibrium is done via the one-block step in Guo et al. (Commun Math Phys 118:31, 1988) for path-space/dynamic statistics. [ABSTRACT FROM AUTHOR]

Three phases of macroscopic domains have been seen for large but finite periodic dimer models; these are known as the frozen, rough and smooth phases. The transition region between the frozen and rough region has received a lot of attention for the last twenty years and recently work has been underway to understand the rough–smooth transition region in the case of the two-periodic Aztec diamond. We compute uniform asymptotics for dimer–dimer correlations of the two-periodic Aztec diamond when the dimers lie in the rough–smooth transition region. These asymptotics rely on a formula found in Chhita and Johansson (Adv Math 294:37–149, 2016) for the inverse Kasteleyn matrix, they also apply to the infinite square grid dimer model with a variable weighting which interpolates between the rough and smooth phase (Kenyon et al. Ann Math (2) 163(3):1019–1056, 2006). In particular, we find that distant dimers initially decay exponentially when the magnetic coordinates are very close to the bounded complementary component of the associated amoebae, they then transition to a power law decay once far enough apart. [ABSTRACT FROM AUTHOR]

We establish the Q Q ~ -systems for the twisted quantum affine algebras that were conjectured in Frenkel and Hernandez (Commun Math Phys 362(2):361–414, 2018). We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations. [ABSTRACT FROM AUTHOR]

It is well-known that the value of the Frobenius–Schur indicator | G | - 1 ∑ g ∈ G χ (g 2) = ± 1 of a real irreducible representation of a finite group G determines which of the two types of real representations it belongs to, i.e. whether it is strictly real or quaternionic. We study the extension to the case when a homomorphism φ : G → Z / 2 Z gives the group algebra C [ G ] the structure of a superalgebra. Namely, we construct of a super version of the Frobenius–Schur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the eight types of irreducible real super representations described in Wall (in J Reine Angew Math 213:187–199, 1963/64. https://doi.org/10.1515/crll.1964.213.187) it belongs to. We also discuss its significance in the context of two-dimensional finite-group gauge theories on pin - surfaces. [ABSTRACT FROM AUTHOR]

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density ρ 0 is discontinuous and the initial velocity u 0 has critical regularity. Assuming that ρ 0 is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity u 0 belongs to the critical homogeneous Besov space B ˙ p , 1 - 1 + 2 / p (R 2) (1 < p < 2) and, in the three-dimensional case, if u 0 is small in B ˙ p , 1 - 1 + 3 / p (R 3) (1 < p < 3) . Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by Zhang (Adv Math 363:107007, 2020) are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Generalizing Frassek et al. (Adv. Math. 401, 108283 (2022). https://doi.org/10.1016/j.aim.2022.108283), we construct a family of SO(2r), Sp(2r), S O (2 r + 1) rational Lax matrices T D (z) , polynomial in the spectral parameter z, parametrized by Λ + -valued divisors D on P 1 . To this end, we provide the RTT realization of the antidominantly shifted extended Drinfeld Yangians of g = so 2 r , sp 2 r , so 2 r + 1 , and of their coproduct homomorphisms. [ABSTRACT FROM AUTHOR]

We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S. Bögli (Commun Math Phys 352:629–639, 2017), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull Lond Math Soc 43:745–750, 2011 and Trans Am Math Soc 370:219–240, 2018) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann Inst H Poincaré Sect A (N.S.) 38:7–13, 1983) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp. [ABSTRACT FROM AUTHOR]

We establish a surface order large deviation estimate for the magnetisation of low temperature ϕ 3 4 . As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the ϕ 3 4 singular stochastic PDE. Our main technical contributions are contour bounds for ϕ 3 4 , which extends 2D results by Glimm et al. (Commun Math Phys 45(3):203–216, 1975). We adapt an argument by Bodineau et al. (J Math Phys 41(3):1033–1098, 2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of ϕ 3 4 whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for ϕ 3 4 developed recently by Barashkov and Gubinelli (Duke Math. J. 169(17):3339–3415, 2020). [ABSTRACT FROM AUTHOR]

We extend both the Hawking–Penrose theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity C 1 . For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for C 1 -metrics (Graf in Commun Math Phys 378(2):1417–1450, 2020). For the second issue, we study geodesic branching and add a further alternative to causal geodesic incompleteness to the theorem, namely a condition of maximal causal non-branching. The genericity condition is re-cast in a distributional form that applies to the current reduced regularity while still being fully compatible with the smooth and C 1 , 1 -settings. In addition, we develop refinements of the comparison techniques used in the proof of the C 1 , 1 -version of the theorem (Graf in Commun Math Phys 360:1009–1042, 2018). The necessary results from low regularity causality theory are collected in an appendix. [ABSTRACT FROM AUTHOR]

We demonstrate that random tensors transforming under rank-5 irreducible representations of O (N) can support melonic large N expansions. Our construction is based on models with sextic (5-simplex) interaction, which generalize previously studied rank-3 models with quartic (tetrahedral) interaction (Benedetti et al. in Commun Math Phys 371:55, 2019. arXiv:1712.00249; Carrozza in JHEP 06:039, 2018. arXiv:1803.02496). Beyond the irreducible character of the representations, our proof relies on recursive bounds derived from a detailed combinatorial analysis of the Feynman graphs. Our results provide further evidence that the melonic limit is a universal feature of irreducible tensor models in arbitrary rank. [ABSTRACT FROM AUTHOR]

Let M be a semifinite von Neumann algebra with a faithful normal semifinite trace τ and let A be an arbitrary von Neumann subalgebra of M . We characterize the class of symmetric ideals E in M such that derivations δ : A → E are necessarily inner, which is a unification and far-reaching extension of the results due to Johnson and Parrott (J Funct Anal 11:39–61, 1972), due to Kaftal and Weiss (J Funct Anal 62:202–220, 1985), and due to Popa (J Funct Anal 71:393–408, 1987). In particular, we show that every derivation from A into the ideal C 0 (M , τ) of all τ -compact operators is inner, establishing a semifinite version of the Johnson–Parrott–Popa Theorem which is different from Popa and Rădulescu (Duke Math J 57(2):485–518, 1988, Theorem 1.1) and contrasts to the example of a non-inner derivation established in Popa and Rădulescu (1988, Theorem 1.2). [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr–Sommerfeld quantization rule applied to the normal form Hamiltonian and also admits an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in Bambusi et al. (Commun Part Differ Equ 45:1–18, 2020) (see also Parnovski and Sobolev in Invent Math 181(3):467–540, 2010) for the particular case of T d . [ABSTRACT FROM AUTHOR]

In Schweigert and Yang (Methods Appl Symmetry Integr Geom, 2021) it was shown how string-net spaces for the Cardy bulk algebra in the Drinfeld center Z (C) of a modular tensor category C give rise to a consistent set of correlators. We extend their results to include open-closed world sheets and allow for more general field algebras, which come in the form of (C | Z (C)) -Cardy algebras. To be more precise, we show that a set of fundamental string-nets with input data from a (C | Z (C)) -Cardy algebra gives rise to a solution of the sewing constraints formulated in Kong et al. (Adv Math 262:604–681, 2014) and that any set of fundamental string-nets solving the sewing constraints determine a (C | Z (C)) -Cardy algebra up to isomorphism. Hence we give an alternative proof of the results in Kong et al. (2014) in terms of string-nets. [ABSTRACT FROM AUTHOR]

We construct an infinite volume spatial random permutation (X , σ) , where X ⊂ R d is locally finite and σ : X → X is a permutation, associated to the formal Hamiltonian H (X , σ) = ∑ x ∈ X ‖ x - σ (x) ‖ 2. The measures are parametrized by the point density ρ and the temperature α . Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (Phys Rev 91:1291–1301, 1953). Let ρ c = ρ c (α) be the critical density for Bose–Einstein condensation in Feynman's representation. Each finite cycle of σ induces a loop of points of X . For ρ ≤ ρ c we define (X , σ) as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (Probab Theory Related Fields 128(4):565–588, 2004). We also construct Gaussian random interlacements, a Poisson process of doubly infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (Ann Math 2 171(3):2039–2087, 2010). For d ≥ 3 and ρ > ρ c we define (X , σ) as the superposition of independent realizations of the Gaussian loop soup at density ρ c and the Gaussian random interlacements at density ρ - ρ c . In either case we call (X , σ) a Gaussian random permutation at density ρ and temperature α . The resulting measure satisfies a Markov property and it is Gibbs for the Hamiltonian H. Its point marginal X has the same distribution as the boson point process introduced by Shirai-Takahashi (J Funct Anal 205(2):414–463, 2003) in the subcritical case, and by Tamura-Ito (J Funct Anal 243(1): 207–231, 2007) in the supercritical case. [ABSTRACT FROM AUTHOR]

Any totally positive (k + m) × n matrix induces a map π + from the positive Grassmannian Gr + (k , n) to the Grassmannian Gr (k , k + m) , whose image is the amplituhedron A n , k , m and is endowed with a top-degree form called the canonical form Ω (A n , k , m) . This construction was introduced by Arkani-Hamed and Trnka (J High Energy Phys 2014(10):30, 2014), where they showed that Ω (A n , k , 4) encodes scattering amplitudes in N = 4 super Yang-Mills theory. One way to compute Ω (A n , k , m) is to subdivide A n , k , m into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of A n , k , 4 . However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes (Billera and Sturmfels in Ann Math 527–549, 1992; Gelfand et al. in Discriminants, resultants, and multidimensional determinants, Birkahuser, Boston, 1994), the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of A n , k , m in order to establish the notion of secondary amplituhedron. For this purpose, we first extend the projection π + to a rational map π : G r (k , n) ⤏ G r (k , k + m) and provide a concrete birational parametrization of the fibers of π . We then use this to explicitly describe a rational top-degree form ω n , k , m (with simple poles) on the fibers and compute Ω (A n , k , m) as a summation of certain residues of ω n , k , m . As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when n - k - 1 = m (even). We show that, in this case, each fiber of π is parametrized by a projective space and its volume form ω n , k , m has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey–Kirwan residue computes Ω (A n , k , m) from the fiber volume form ω n , k , m . In particular, we give conceptual proofs of the statements of Ferro et al. (J Phys A Math Theor 52(4):045201, 2018). Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes. [ABSTRACT FROM AUTHOR]