We prove a distributional limit theorem conjectured in Clark (J Stat Phys 174(6):1372–1403, 2019) for partition functions defining models of directed polymers on diamond hierarchical graphs with disorder variables placed at the graphical edges. The limiting regime involves a joint scaling in which the number of hierarchical layers, \(n\in {\mathbb {N}}\), of the graphs grows as the inverse temperature

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of \({\mathbb {R}}^2\). Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegő type inequality as well as a new

Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state \(p=k\varrho \), \(k>0\), and subject to Newtonian gravity. We rigorously prove the existence of such a Larson–Penston

We introduce a task that we call partial decoupling, in which a bipartite quantum state is transformed by a unitary operation on one of the two subsystems and then is subject to the action of a quantum channel. We assume that the subsystem is decomposed into a direct-sum-product form, which often appears in the context of quantum information theory. The unitary is chosen at random from the set of unitaries

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity \(\Omega \), such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization

We consider quantum spins with \(S\ge 1\), and two-body interactions with \(O(2S+1)\) symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for S sufficiently large. We also prove the existence of a gap for excitations.

We study the \(\mathfrak {gl}_{1|1}\) supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation \(\mathfrak {gl}_{1|1}\) Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial

We investigate the statistical properties of translation invariant random fields (including point processes) on Euclidean spaces (or lattices) under constraints on their spectrum or structure function. An important class of models that motivate our study are hyperuniform and stealthy hyperuniform systems, which are characterised by the vanishing of the structure function at the origin (resp., vanishing

Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers \(U^{\otimes t}\) of all unitaries \(U\in U(d)\) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object

We construct two types of multi-layer quantum graphs (Schrödinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. Each component contributes a set of bands to the spectrum of the graph

We develop for the first time a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential \(\varphi \) (in the sense of Liverani–Saussol–Vaienti), we prove there exists a unique random conformal measure \(\nu _\varphi \) and unique random equilibrium

In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact Kähler manifold. The starting point of our study is Schumacher–Toma/Biswas–Schumacher’s curvature formulas for Weil–Petersson-type metrics, in Sect. 2, we give some applications of their formulas on the geometric properties of the base

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate

We investigate a class of local quantum circuits on chains of d-level systems (qudits) that share the so-called ‘dual unitarity’ property. In essence, the latter property implies that these systems generate unitary dynamics not only when propagating in time, but also when propagating in space. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value

Let A and B be local operators in Hamiltonian quantum systems with N degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm \(\Vert [A(t),B]\Vert \) is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian’s factor graph. Our bounds sharpen existing Lieb–Robinson bounds by removing extraneous

In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein–Hilbert and of the Palatini–Cartan theories in every space-time dimension greater than two. In the former case one obtains a BV theory for the first-order formulation of Einstein–Hilbert theory, in the latter a BV theory for Palatini–Cartan theory with a partial implementation of the torsion-free

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected

We analyze the propagation of wave packets through general Hamiltonian systems presenting codimension one eigenvalue crossings. The class of time-dependent Hamiltonians we consider is of general pseudodifferential form with subquadratic growth. It comprises Schrödinger operators with matrix-valued potential, as they occur in quantum molecular dynamics, but also covers matrix-valued models of solid

We show how to use Gromov-Witten invariants to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau manifolds X over Hirzebruch surfaces. To determine the representations of these matter multiplets under the gauge algebra \({\mathfrak {g}}\), we use toric methods to embed the weight lattice of \({\mathfrak {g}}\) into the integer homology lattice of X. We then

For the classification of SPT phases, defining an index is a central problem. In the famous paper (Pollmann et al. Phys Rev B 81:064439, 2010), Pollmann, Tuner, Berg, and Oshikawa introduced \({{\mathbb {Z}}}_2\)-indices for injective matrix products states which have either \({{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2\) dihedral group (of \(\pi \)-rotations about x, y, and z-axes) symmetry, time-reversal

We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner–Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over \({\mathbb {C}}\), and to some extent also over \({\mathbb {R}}\), up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are \(\mathfrak {sl}_2\), \(\mathfrak {sp}_4\) and \(\mathfrak {sp}_{10}\). Among these, each admits exactly

We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction

We consider 1d random Hermitian \(N\times N\) block band matrices consisting of \(W\times W\) random Gaussian blocks (parametrized by \(j,k \in \Lambda =[1,n]\cap \mathbb {Z}\), \(N=nW\)) with a fixed entry’s variance \(J_{jk}=W^{-1}(\delta _{j,k}+\beta \Delta _{j,k})\) in each block. Considering the limit \(W, n\rightarrow \infty \), we prove that the behaviour of the second correlation function of

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a weight \(r>0\). We give a variational principle for limit shapes in this setting, and show that the resulting Euler–Lagrange equation can be integrated, giving limit shapes explicitly parameterized by analytic functions.

This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound \(\Vert u\Vert _{L^{\infty }_{t}L^{3,\infty }_{x}}\le M\). Namely, we show that if \(T^*\) is a first blow-up time and \((0,T^*)\) is a singular point then $$\begin{aligned} \Vert

Let G be a connected semi-simple Lie group with torsion-free fundamental group. We show that the twisted equivariant KK-theory \(KK_{\bullet }^{G}(G/K, \tau _G^G)\) of G has a ring structure induced from the renowned ring structure of the twisted equivariant K-theory \(K^{\bullet }_{K}(K, \tau _K^K)\) of a maximal compact subgroup K. We give a geometric description of representatives in \(KK_{\bullet

Supersymmetric D-branes supported on the complex two-dimensional base S of the local Calabi–Yau threefold \(K_S\) are described by semi-stable coherent sheaves on S. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson–Thomas invariants) coincide with the Vafa–Witten invariants of S (which encode the Betti numbers of the moduli space of semi-stable sheaves)

Let G be a countable discrete group and consider a nonsingular Bernoulli shift action \( G \curvearrowright \prod _{g\in G }(\{0,1\},\mu _g)\) with two base points. We prove the first rigidity result for Bernoulli shift actions that are not measure preserving, by proving solidity for certain non-singular Bernoulli actions, making use of a new boundary associated with such Bernoulli actions. This generalizes

We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states \(\rho ^{\otimes n}\) against convex combinations of quantum states \(\sigma ^{\otimes n}\) can be written as a regularized quantum relative entropy formula. We prove that

It is shown that for any unital infinite dimensional simple AF algebra A and for any lower bounded closed set K of real numbers containing zero there is a flow on A such that the set inverse temperatures is exactly K.

The asymptotic expansion of quantum knot invariants in complex Chern–Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of q-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear q-difference equations. We further

We consider the Vlasov–Poisson system with repulsive interactions. For initial data a small, radial, absolutely continuous perturbation of a point charge, we show that the solution is global and disperses to infinity via a modified scattering along trajectories of the linearized flow. This is done by an exact integration of the linearized equation, followed by the analysis of the perturbed Hamiltonian

We study a family of unbounded solutions to the Korteweg–de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg–de Vries solutions are well-defined for \(x>0\), but not for \(x<0\), where the solutions behave like \(\frac{x}{2t}\)

We study a class of Hamilton–Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special

We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb–Schultz–Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrödinger operators each of which has a simple well potential. Focusing on the cases where the two underlying classical periodic trajectories cross to each other, we give Bohr–Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal

Let Z be a principal circle bundle over a base manifold M equipped with an integral closed 3-form H called the flux. Let \({\widehat{Z}}\) be the T-dual circle bundle over M with flux \({\widehat{H}}\). Han and Mathai recently constructed the \(\mathbb {Z}_2\)-graded space of exotic differential forms \(\mathcal {A}^{\bar{k}}({\widehat{Z}})\). It has an additional \(\mathbb {Z}\)-grading such that

Recent understanding of the thermodynamics of small-scale systems have enabled the characterization of the thermodynamic requirements of implementing quantum processes for fixed input states. Here, we extend these results to construct optimal universal implementations of a given process, that is, implementations that are accurate for any possible input state even after many independent and identically

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically

We study the Langevin dynamics of a U(1) lattice gauge theory on the two-dimensional torus, and prove that they converge for short time in a suitable gauge to a system of stochastic PDEs driven by space-time white noises. We fix gauge via a DeTurck trick. This also yields convergence of some gauge invariant observables on a short time interval. The proof relies on a discrete version of the theory of

For the two-sided Bernoulli initial condition with density \(\rho _-\) (resp. \(\rho _+\)) to the left (resp. to the right), we study the distribution of a tagged particle in the one dimensional symmetric simple exclusion process. We obtain a formula for the moment generating function of the associated current in terms of a Fredholm determinant. Our arguments are based on a combination of techniques

We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere \(\hat{\mathbb {C}}.\) We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values \(z\in \hat{\mathbb

We study bundle gerbes on manifolds M that carry an action of a connected Lie group G. We show that these data give rise to a smooth 2-group extension of G by the smooth 2-group of hermitean line bundles on M. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach

Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels,

A characterization of the Kadomtsev–Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic–geometrical solutions

In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter

We investigate the large time behavior of compactly supported smooth solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. Relaxation rates are obtained for initial data which are close to a steady state in a suitable sense. The proof uses the Lagrangian coordinates. Our method is to establish and exploit

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold

We prove global existence backwards from the scattering data posed at infinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying the weak null condition. The asymptotics of the solutions to the Maxwell Klein Gordon equations in Lorenz gauge were shown to be wave like at null infinity and homogeneous towards timelike infinity in Candy et al. (Commun Math Phys 367(2):683–716, 2019) and

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized m-Bakry–Émery Ricci curvature, in which m is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound, as well as control on the Bakry–Émery vector field, the

In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator \(-\Delta +V(x)\) are raised to the power \(\kappa \) is never given by the one-bound state case when \(\kappa >\max (0,2-d/2)\) in space dimension \(d\ge 1\). When in addition \(\kappa

We use the highly symmetric Stenzel Calabi–Yau structure on \(T^{\star }S^{4}\) as a testing ground for the relationship between the Spin(7) instanton and Hermitian–Yang–Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with

We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller,

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington–Kirkpatrick model. In the seminal work, Bolthausen (Commun Math Phys 325(1):333–366, 2014) introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida–Thouless transition line. However, it was unclear if this

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size \(\varepsilon > 0\), a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least \(O(\varepsilon ^{-2})\). To the best of

We show, using semiclassical measures and unstable derivatives, that a smooth vector field X generating a contact Anosov flow on a 3-dimensional manifold \(\mathcal {M}\) has only finitely many Ruelle resonances in the vertical strips \(\{ s\in \mathbb {C}\ |\ \mathrm{Re}(s)\in [-\nu _{\min }+\varepsilon ,-\frac{1}{2}\nu _{\max }-\varepsilon ]\cup [-\frac{1}{2}\nu _{\min }+\varepsilon ,0]\}\) for all

By introducing extrinsic noise as well as intrinsic uncertainty into a network with stochastic events, this paper studies the dynamics of the resulting Markov random network and characterizes a novel phenomenon of intermittent synchronization and desynchronization that is due to an interplay of the two forms of randomness in the system. On a finite state space and in discrete time, the network allows

We prove the stability of a planar contact discontinuity without shear, a family of special discontinuous solutions for the three-dimensional full Euler system, in the class of vanishing dissipation limits of the corresponding Navier–Stokes–Fourier system. We also show that solutions of the Navier–Stokes–Fourier system converge to the planar contact discontinuity when the initial datum converges to