Communications in Mathematical Physics最新文献

{"title":"Trained Quantum Neural Networks are Gaussian Processes","authors":"Filippo Girardi,&nbsp;Giacomo De Palma","doi":"10.1007/s00220-025-05238-0","DOIUrl":"10.1007/s00220-025-05238-0","url":null,"abstract":"<div><p>We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges in distribution to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. We prove that, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143749231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotics of Helmholtz–Kirchhoff Point-Vortices in the Phase Space","authors":"Chanwoo Kim,&nbsp;Trinh T. Nguyen","doi":"10.1007/s00220-025-05264-y","DOIUrl":"10.1007/s00220-025-05264-y","url":null,"abstract":"<div><p>A rigorous derivation of point vortex systems from kinetic equations has been a challenging open problem, due to singular layers in the inviscid limit, giving a large velocity gradient in the Boltzmann equations. In this paper, we derive the Helmholtz–Kirchhoff point-vortex system from the hydrodynamic limits of the Boltzmann equations. We construct Boltzmann solutions by the Hilbert-type expansion associated to the point vortices solutions of the 2D Navier–Stokes equations. We give a precise pointwise estimate for the solution of the Boltzmann equations with small Strouhal number and Knudsen number.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143749232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Integrable Hierarchies and F-Manifolds with Compatible Connection","authors":"Paolo Lorenzoni,&nbsp;Sara Perletti,&nbsp;Karoline van Gemst","doi":"10.1007/s00220-025-05262-0","DOIUrl":"10.1007/s00220-025-05262-0","url":null,"abstract":"<div><p>We study the geometry of integrable systems of hydrodynamic type of the form <span>(w_t=Xcirc w_x)</span> where <span>(circ )</span> is the product of a regular F-manifold. In the first part of the paper, we present a general construction of a connection compatible with the F-manifold structure starting from a frame of vector fields defining commuting flows of hydrodynamic type. In the second part of the paper, using this construction, we study regular F-manifolds with compatible connection and Euler vector field, <span>((nabla ,circ ,e,E))</span>, associated with integrable hierarchies obtained from the solutions of the equation <span>(dcdot d_L ,a_0=0)</span> where <span>(L=Ecirc )</span>. In particular, we show that <i>n</i>-dimensional F-manifolds associated to regular operators <i>L</i> are classified by <i>n</i> arbitrary functions of a single variable. Moreover, we show that flat connections <span>(nabla )</span> correspond to linear solutions <span>(a_0)</span>.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-025-05262-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143749233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"How Long are the Arms in DBM?","authors":"Ilya Losev,&nbsp;Stanislav Smirnov","doi":"10.1007/s00220-025-05276-8","DOIUrl":"10.1007/s00220-025-05276-8","url":null,"abstract":"<div><p>Diffusion limited aggregation and its generalization, dielectric-breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten’s estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in <span>(mathbb {Z}^2)</span> and <span>(mathbb {Z}^3)</span>. The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten’s, our argument provides a new proof for Kesten’s result both in <span>(mathbb {Z}^2)</span> and <span>(mathbb {Z}^3)</span>.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-025-05276-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143749234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Zero-Background Solitons of the Sharp-Line Maxwell–Bloch Equations","authors":"Sitai Li","doi":"10.1007/s00220-025-05281-x","DOIUrl":"10.1007/s00220-025-05281-x","url":null,"abstract":"<div><p>This work is devoted to systematically study general <i>N</i>-soliton solutions possibly containing multiple degenerate soliton groups (DSGs), in the context of the sharp-line Maxwell–Bloch equations with a zero background. We also show that results can be readily migrated to other integrable systems, with the same non-self-adjoint Zakharov–Shabat scattering problem or alike. Results for the focusing nonlinear Schrödinger equation and the complex modified Korteweg–De Vries equation are obtained as explicit examples for demonstrative purposes. A DSG is a localized coherent nonlinear traveling-wave structure, comprised of inseparable solitons with identical velocities. Hence, DSGs are generalizations of single solitons (considered as 1-DSGs), and form fundamental building blocks of solutions of many integrable systems. We provide an explicit formula for an <i>N</i>-DSG and its center. With the help of the Deift–Zhou’s nonlinear steepest descent method, we prove the localization of DSGs, and calculate the long-time asymptotics for an arbitrary <i>N</i>-soliton solutions. It is shown that the solution becomes a linear combination of multiple DSGs in the distant past and future, with explicit formulæ for the asymptotic phase shift for each DSG. Other generalizations of a single soliton are also discussed, such as <i>N</i>th-order solitons and soliton gases. We prove that every <i>N</i>th-order soliton can be obtained by fusion of eigenvalues of <i>N</i>-soliton solutions, with proper rescalings of norming constants, and demonstrate that soliton-gas solution can be considered as limits of <i>N</i>-soliton solutions as <span>(Nrightarrow +infty )</span>.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143749215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Correction to: Feigin–Frenkel–Hernandez Opers and the (QQ-)System","authors":"Davide Masoero,&nbsp;Andrea Raimondo","doi":"10.1007/s00220-025-05286-6","DOIUrl":"10.1007/s00220-025-05286-6","url":null,"abstract":"","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Metastability in Glauber Dynamics for Heavy-Tailed Spin Glasses","authors":"Reza Gheissari,&nbsp;Curtis Grant","doi":"10.1007/s00220-025-05265-x","DOIUrl":"10.1007/s00220-025-05265-x","url":null,"abstract":"<div><p>We study the Glauber dynamics for heavy-tailed spin glasses, in which the couplings are in the domain of attraction of an <span>(alpha )</span>-stable law for <span>(alpha in (0,1))</span>. We show a sharp description of metastability on exponential timescales, in a form that is believed to hold for Glauber/Langevin dynamics for many mean-field spin glass models, but only known rigorously for the Random Energy Models. Namely, we establish a decomposition of the state space into sub-exponentially many wells, and show that the projection of the Glauber dynamics onto which well it resides in, asymptotically behaves like a Markov chain on wells with certain explicit transition rates. In particular, mixing inside wells occurs on much shorter timescales than transit times between wells, and the law of the next well the Glauber dynamics will fall into depends only on which well it currently resides in, not its full configuration. We can deduce consequences like an exact expression for the two-time autocorrelation functions that appear in the activated aging literature.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143655218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Random Matrix Model Towards the Quantum Chaos Transition Conjecture","authors":"Bertrand Stone,&nbsp;Fan Yang,&nbsp;Jun Yin","doi":"10.1007/s00220-025-05275-9","DOIUrl":"10.1007/s00220-025-05275-9","url":null,"abstract":"<div><p>Consider <i>D</i> random systems that are modeled by independent <span>(Ntimes N)</span> complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix <i>A</i>. We prove that in the asymptotic limit <span>(Nrightarrow infty )</span>, the whole system exhibits a quantum chaos transition when the interaction strength <span>(Vert AVert _{{textrm{HS}}})</span> varies. Specifically, when <span>(Vert AVert _{{textrm{HS}}}ge N^{{varepsilon }})</span>, we prove that the bulk eigenvalue statistics match those of a <span>(DNtimes DN)</span> GUE asymptotically and each bulk eigenvector is approximately equally distributed among the <i>D</i> subsystems with probability <span>(1-textrm{o}(1))</span>. These phenomena indicate quantum chaos of the whole system. In contrast, when <span>(Vert AVert _{{textrm{HS}}}le N^{-{varepsilon }})</span>, we show that the system is integrable: the bulk eigenvalue statistics behave like <i>D</i> independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. In particular, if we take <span>(Drightarrow infty )</span> after the <span>(Nrightarrow infty )</span> limit, the bulk statistics converge to a Poisson point process under the <i>DN</i> scaling.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143655283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Universality of Poisson–Dirichlet Law for Log-Correlated Gaussian Fields via Level Set Statistics","authors":"Shirshendu Ganguly,&nbsp;Kyeongsik Nam","doi":"10.1007/s00220-025-05270-0","DOIUrl":"10.1007/s00220-025-05270-0","url":null,"abstract":"<div><p>Many low temperature disordered systems are expected to exhibit Poisson–Dirichlet (PD) statistics. In this paper, we focus on the case when the underlying disorder is a logarithmically correlated Gaussian process <span>(phi _N)</span> on the box <span>([0,N]^dsubset mathbb {Z}^d)</span>. Canonical examples include branching random walk, <span>(*)</span>-scale invariant fields, with the central example being the two dimensional Gaussian free field (GFF), a universal scaling limit of a wide range of statistical mechanics models. The corresponding Gibbs measure obtained by exponentiating <span>(beta )</span> (inverse temperature) times <span>(phi _N)</span> is a discrete version of the Gaussian multiplicative chaos (GMC) famously constructed by Kahane (Ann Sci Math Québec 9(2): 105–150, 1985). In the low temperature or supercritical regime, i.e., <span>(beta )</span> larger than a critical <span>(beta _c,)</span> the GMC is expected to exhibit atomic behavior on suitable renormalization, dictated by the extremal statistics or near maximum values of <span>(phi _N)</span>. Moreover, it is predicted going back to a conjecture made in 2001 in Carpentier and Le Doussal (Phys Rev E 63(2): 026110, 2001), that the weights of this atomic GMC has a PD distribution. In a series of works culminating in Biskup and Louidor (Adv Math 330, 589–687, 2018), Biskup and Louidor carried out a comprehensive study of the near maxima of the 2D GFF, and established the conjectured PD behavior throughout the super-critical regime (<span>(beta &gt; 2)</span>). In another direction Ding et al. (Ann Probab 5(6A), 3886–3928, 2017), established universal behavior of the maximum for a general class of log-correlated Gaussian fields. In this paper we continue this program simply under the assumption of log-correlation and nothing further. We prove that the GMC concentrates on an <i>O</i>(1) neighborhood of the local extrema and the PD prediction made in Carpentier and Le Doussal (Phys Rev E 63(2): 026110, 2001) holds, in any dimension <i>d</i>, throughout the supercritical regime <span>(beta &gt; sqrt{2d})</span>, significantly generalizing past results. While many of the arguments for the GFF make use of the powerful Gibbs–Markov property, in absence of any Markovian structure for general Gaussian fields, we develop and use as our key input a sharp estimate of the size of level sets, a result we believe could have other applications.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143655313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Shuffle Theorems and Sandpiles","authors":"Michele D’Adderio,&nbsp;Mark Dukes,&nbsp;Alessandro Iraci,&nbsp;Alexander Lazar,&nbsp;Yvan Le Borgne,&nbsp;Anna Vanden Wyngaerd","doi":"10.1007/s00220-025-05233-5","DOIUrl":"10.1007/s00220-025-05233-5","url":null,"abstract":"<div><p>We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs <span>({widehat{G}}_{mu ,nu })</span>, which we call <i>clique-independent</i> graphs, indexed by two compositions <span>(mu )</span> and <span>(nu )</span>. Moreover, we define a <i>delay</i> statistic on these configurations, and we show that, together with the usual <i>level</i> statistic, it can be used to provide a new combinatorial interpretation of the celebrated <i>shuffle theorem</i> of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials <span>(langle nabla e_n, e_mu h_nu rangle )</span> in terms of these configurations.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 4","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143655315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}