Comm Math Phys Research Articles

A Nekhoroshev-type theorem for the nonlinear Schrödinger equations on the d-dimensional torus

We prove a Nekhoroshev type long time stability result for the following nonlinear Schrödinger (NLS) equations on the d-dimensional torus i u t = − △ u + V ∗ u + ∂ F ( | u | 2 ) ∂ u ¯ ( u , u ¯ ) , x ∈ T d , t ∈ R , where V is a smooth convolution potential, F ( z ) is a real-valued polynomial function in z satisfying F ( 0 ) = F ′ ( 0 ) = 0 . More precisely, it is proved that in modified Sobolev space (see (2)) if the norm of initial datum is smaller than ε ( 0 < ε ≪ 1 ) , then the corresponding solution is bounded by 2 ε over time-intervals of length e e 1 ε . The result generalizes the 1-dimensional case in L. Biasco, J.E. Massetti, M. Procesi. [An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS. Comm Math Phys. 2020;375(3):2089–2153] to d-dimensional NLS equations.

Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach

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.

Ke Li’s Lemma for Quantum Hypothesis Testing in General Von Neumann Algebras

A lemma stated by Li (Ann Statist 42:171-189, 2014) has been used in, for example, Nilanjana et al. (J Math Phys 57:062207, 2016), Kaur et al. (Phys Rev A 96:062318, 2017), Rouzé, Datta (IEEE Trans Inform Theory 64:593–612, 2018), Tomamichel, Tan (Comm Math Phys 338:103-137, 2015) and Wilde et al. (IEEE Trans Inform Theory 63:1792-1817, 2017) for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here, we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields a new proof of quantum Stein’s lemma with slightly weaker assumption, as well as immediate generalizations of its second-order asymptotics, for example the main results in Nilanjana et al. (J Math Phys 57:062207, 2016) and Li (Ann Statist 42:171-189, 2014).

Random-field random surfaces

We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued random-field random surfaces of the $$\nabla \phi $$ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions $$1\le d\le 2$$ and localizes in dimensions $$d\ge 3$$ . (ii) The surface delocalizes in dimensions $$1\le d\le 4$$ and localizes in dimensions $$d\ge 5$$ . It is further shown that for the integer-valued random-field Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $$d=1,2$$ and localizes in dimensions $$d\ge 3$$ . (ii) The surface delocalizes in dimensions $$d=1,2$$ . (iii) The surface localizes in dimensions $$d\ge 3$$ at low temperature and weak disorder strength. The behavior in dimensions $$d\ge 3$$ at high temperature or strong disorder is left open. The proofs rely on several tools: Explicit identities satisfied by the expectation of the random surface, the Efron–Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn (Comm Math Phys 185(1): 1-36, 1997) and the Nash–Aronson estimate.

A family of non-modular covariant AQFTs

Based on the construction provided in our paper “Covariant homogeneous nets of standard subspaces”, Comm Math Phys 386:305–358, (2021), we construct non-modular covariant one-particle nets on the two-dimensional de Sitter spacetime and on the three-dimensional Minkowski space.

High Dimensional Normality of Noisy Eigenvectors

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) (Bourgade and Yau in Comm Math Phys, 2013) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow first introduced in Bourgade and Yau (Comm Math Phys, 2013) to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for Lévy matrices when the eigenvectors correspond to small energy levels.

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

Density of Small Singular Values of the Shifted Real Ginibre Ensemble

We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold.

Modularity of Bershadsky–Polyakov minimal models

The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with mathfrak {sl}_3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level mathsf {k} is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with mathfrak {sl}_2, except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s mathsf {W}_3 algebras.

Mixing solutions for the Muskat problem with variable speed

We provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).

Ergodization time for linear flows on tori via geometry of numbers

In this paper, we give a new, short, simple, and geometric proof of the optimal ergodization time for linear flows on tori. This result was first obtained by Bourgain et al. (Comm Math Phys 190:491–508, 1998) using Fourier analysis. Our proof uses geometry of numbers: it follows trivially from a Diophantine duality that was established by Bounemoura and Fischler (Math 275:1135–1167, 2013).

Inequalities of Simons type and gaps for Yang–Mills fields

In this paper, we establish an inequality of Simons type for Yang–Mills fields, and obtain a gap property, which generalize the results obtained in Bourguignon and Lawson (Comm Math Phys 79(2):189–230, 1981).

Weyl modules and $$q$$ q -Whittaker functions

Let \(G\) be a semi-simple simply connected group over \(\mathbb {C}\). Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the \(q\)-Toda integrable system obtained by quantum group version of the Kostant–Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9–25, 1999, Sevostyanov in Commun Math Phys 204:1–16, 1999) to define the notion of \(q\)-Whittaker functions \(\varPsi _{\check{\lambda }}(q,z)\). This is a family of invariant polynomials on the maximal torus \(T\subset G\) (here \(z\in T\)) depending on a dominant weight \(\check{\lambda }\) of \(G\) whose coefficients are rational functions in a variable \(q\in \mathbb {C}^*\). For a conjecturally the same (but a priori different) definition of the \(q\)-Toda system these functions were studied by Ion (Duke Math J 116:1–16, 2003) and by Cherednik (Int Math Res Notices 20:3793–3842, 2009) [we shall denote the \(q\)-Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by \(\varPsi '_{\check{\lambda }}(q,z)\)]. For \(G=SL(N)\) these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97–119, 2010; Comm Math Phys 294:121–143, 2010; Lett Math Phys 97:1–24, 2011). We show that when \(G\) is simply laced, the function \(\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}\) (here \(I\) denotes the set of vertices of the Dynkin diagram of \(G\)) is equal to the character of a certain finite-dimensional \(G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*\)-module \(D(\check{\lambda })\) (the Demazure module). When \(G\) is not simply laced a twisted version of the above statement holds. This result is known for \(\varPsi _{\check{\lambda }}\) replaced by \(\varPsi '_{\check{\lambda }}\) (cf. Sanderson in J Algebraic Combin 11:269–275, 2000 and Ion in Duke Math J 116:1–16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum \(K\)-theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269–275, 2000) and Ion (Duke Math J 116:1–16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules \(D(\check{\lambda })]\).

A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment

We consider a random walk on $$\mathbb{Z }^d, d\ge 2$$ , in an i.i.d. balanced random environment, that is a random walk for which the probability to jump from $$x\in \mathbb{Z }^d$$ to nearest neighbor $$x+e$$ is the same as to nearest neighbor $$x-e$$ . Assuming that the environment is genuinely $$d$$ -dimensional and balanced we show a quenched invariance principle: for $$P$$ almost every environment, the diffusively rescaled random walk converges to a Brownian motion with deterministic non-degenerate diffusion matrix. Within the i.i.d. setting, our result extend both Lawler’s uniformly elliptic result (Comm Math Phys, 87(1), pp 81–87, 1982/1983) and Guo and Zeitouni’s elliptic result (to appear in PTRF, 2010) to the general (non elliptic) case. Our proof is based on analytic methods and percolation arguments.

Class A spacetimes

We introduce class A spacetimes, i.e. compact vicious spacetimes (M, g) such that the Abelian cover \({(\overline{M}, \overline{g})}\) is globally hyperbolic. We study the main properties of class A spacetimes using methods similar to those introduced in Sullivan (Invent Math 36:225–255, 1976) and Burago (Adv Sov Math 9:205–210, 1992). As a consequence we are able to characterize manifolds admitting class A metrics completely as mapping tori. Further we show that the notion of class A spacetime is equivalent to that of SCTP (spacially compact time-periodic) spacetimes as introduced in Galloway (Comm Math Phys 96:423–429, 1984). The set of class A spacetimes is shown to be open in the C 0-topology on the set of Lorentzian metrics. As an application we prove a coarse Lipschitz property for the time separation of the Abelian cover. This coarse Lipschitz property is an essential part in the study of Aubry-Mather theory in Lorentzian geometry.

Lorentz Gas with Thermostatted Walls

In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity (Chernov and Dolgopyat in J Am Math Soc 22:821–858, 2009; Phys Rev Lett 99, paper 030601, 2007). To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle’s speed to be constant. Then an electric current sets in and one can prove Ohm’s law and the Einstein relation (Chernov and Dolgopyat in Russian Math Surv 64:73–124, 2009; Chernov et al. Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993). However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the ‘walls’). We rederive all physically interesting facts proven for the Gaussian thermostat in Chernov, Dolgopyat (Russian Math Surv 64:73–124, 2009) and Chernov et al. (Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993), including Ohm’s law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.

Nonlinear stability for advective systems

We consider partial differential equations of “advective” type, in the sense defined in Shvydkoy (Comm Math Phys 265:507–545, 2006). We establish rigorous results that allow to infer nonlinear stability for small perturbations from linearized stability results.

Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335–358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551–582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267–1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdős-Rényi random graphs.

Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris’ theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such “asymptotic couplings” were central to (Mattingly and Sinai in Comm Math Phys 219(3):523–565, 2001; Mattingly in Comm Math Phys 230(3):461–462, 2002; Hairer in Prob Theory Relat Field 124:345–380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553–582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris’ celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are “small” (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a “small set” by the much weaker notion of a “d-small set,” which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris’ theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.

Symmetry and monotonicity of least energy solutions

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Maris (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brezis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincare Anal Non Lineaire 1:223–283, 1984).