Mathematical Physics
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Showing new listings for Thursday, 17 April 2025
- [1] arXiv:2504.11532 [pdf, html, other]
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Title: Infinite Stability in Disordered SystemsSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Superconductivity (cond-mat.supr-con)
In quenched disordered systems, the existence of ordering is generally believed to be only possible in the weak disorder regime (disregarding models of spin-glass type). In particular, sufficiently large random fields is expected to prohibit any finite temperature ordering. Here, we prove that this is not necessarily true, and show rigorously that for physically relevant systems in $\mathbb{Z}^d$ with $d\ge 3$, disorder can induce ordering that is \textit{infinitely stable}, in the sense that (1) there exists ordering at arbitrarily large disorder strength and (2) the transition temperature is asymptotically nonzero in the limit of infinite disorder. Analogous results can hold in 2 dimensions provided that the underlying graph is non-planar (e.g., $\mathbb{Z}^2$ sites with nearest and next-nearest neighbor interactions).
- [2] arXiv:2504.11753 [pdf, html, other]
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Title: The $L^p$-boundedness of wave operators for 4-th order Schrödinger operators on $\mathbb{R}^2$, ISubjects: Mathematical Physics (math-ph)
We prove that high energy parts of wave operators for fourth order Schrödinger operators $H=\Delta^2 + V(x)$ in $\mathbb{R}^2$ are bounded in $L^p(\mathbb{R}^2)$ for $p\in(1,\infty)$.
- [3] arXiv:2504.12120 [pdf, html, other]
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Title: Logarithmic Spectral Distribution of a non-Hermitian $β$-EnsembleComments: 47 pages, 9 figuresSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR)
We introduce a non-Hermitian $\beta$-ensemble and determine its spectral density in the limit of large $\beta$ and large matrix size $n$. The ensemble is given by a general tridiagonal complex random matrix of normal and chi-distributed random variables, extending previous work of two of the authors. The joint distribution of eigenvalues contains a Vandermonde determinant to the power $\beta$ and a residual coupling to the eigenvectors. A tool in the computation of the limiting spectral density is a single characteristic polynomial for centred tridiagonal Jacobi matrices, for which we explicitly determine the coefficients in terms of its matrix elements. In the low temperature limit $\beta\gg1$ our ensemble reduces to such a centred matrix with vanishing diagonal. A general theorem from free probability based on the variance of the coefficients of the characteristic polynomial allows us to obtain the spectral density when additionally taking the large-$n$ limit. It is rotationally invariant on a compact disc, given by the logarithm of the radius plus a constant. The same density is obtained when starting form a tridiagonal complex symmetric ensemble, which thus plays a special role. Extensive numerical simulations confirm our analytical results and put this and the previously studied ensemble in the context of the pseudospectrum.
New submissions (showing 3 of 3 entries)
- [4] arXiv:2504.11471 (cross-list from gr-qc) [pdf, html, other]
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Title: Vaidya-Reissner-Nordström Extension On the White-hole Region and the White Hole Evaporation DynamicComments: 30 pages, 2 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
We develop an analytic model that extends classical white hole geometry by incorporating both radiative dynamics and electric charge. Starting from a maximal analytic extension of the Schwarzschild white hole via Kruskal Szekeres coordinates, we introduce a time dependent mass function, representative of outgoing null dust to model evaporation. Building on this foundation, the study then integrates the Reissner-Nordström framework to obtain a dynamic, charged white hole solution in double null coordinates. In the resulting Vaidya Reissner Nordström metric, both the Bondi mass and the associated charge decrease monotonically with retarded time, capturing the interplay of radiation and electromagnetic effects. Detailed analysis of horizon behavior reveals how mass loss and charge shedding modify the causal structure, ensuring that energy conditions are preserved and cosmic censorship is maintained.
- [5] arXiv:2504.11525 (cross-list from quant-ph) [pdf, html, other]
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Title: Entangled Subspaces through Algebraic GeometryComments: 23 pages. Your comments are more than welcomeSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
We propose an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Specifically, our method employs a modified Veronese embedding, restricted to the conic, to define subspaces within the symmetric part of the Hilbert space. By utilizing this technique, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), enabling the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). In multiqudit systems, we determine the maximum achievable dimension of a symmetric GES and demonstrate its realization through this construction. Furthermore, we systematically investigate the transition from the conventional Veronese embedding to the modified one by imposing various constraints on the affine coordinates, which, in turn, increases the CES dimension while reducing that of the GES.
- [6] arXiv:2504.11533 (cross-list from hep-th) [pdf, html, other]
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Title: Self-duality and the Holomorphic Ansatz in Generalized BPS Skyrme ModelComments: 32 pages, 1 figureSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We propose a generalization of the BPS Skyrme model for simple compact Lie groups $G$ that leads to Hermitian symmetric spaces. In such a theory, the Skyrme field takes its values in $G$, while the remaining fields correspond to the entries of a symmetric, positive, and invertible $\dim G \times \dim G$-dimensional matrix $h$. We also use the holomorphic map ansatz between $S^2 \rightarrow G/H \times U(1)$ to study the self-dual sector of the theory, which generalizes the holomorphic ansatz between $S^2 \rightarrow CP^N$. This ansatz is constructed using the fact that stable harmonic maps of the two $S^2$ spheres for compact Hermitian symmetric spaces are holomorphic or anti-holomorphic. Apart from some special cases, the self-duality equations do not fix the matrix $h$ entirely in terms of the Skyrme field, which is completely free, as it happens in the original self-dual Skyrme model for $G=SU(2)$. In general, the freedom of the $h$ fields tend to grow with the dimension of $G$. The holomorphic ansatz enable us to construct an infinite number of exact self-dual Skyrmions for each integer value of the topological charge and for each value of $N \geq 1$, in case of the $CP^N$, and for each values of $p,\,q\geq 1$ in case of $SU(p+q)/SU(p)\otimes SU(q)\otimes U(1)$.
- [7] arXiv:2504.11590 (cross-list from math.OC) [pdf, html, other]
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Title: Approximating a matrix as the square of a skew-symmetric matrix, with application to estimating angular velocity from acceleration dataSubjects: Optimization and Control (math.OC); Mathematical Physics (math-ph)
In this paper we study the problem of finding the best approximation of a real square matrix by a matrix that can be represented as the square of a real, skew-symmetric matrix. This problem is important in the design of robust numerical algorithms aimed at estimating rigid body kinematics from multiple accelerometer measurements. We give a constructive proof for the existence of a best approximant in the Frobenius norm. We demonstrate the construction with some small examples, and we showcase the practical importance of this work to the problem of determining the angular velocity of a rotating rigid body from its acceleration measurements.
- [8] arXiv:2504.11747 (cross-list from quant-ph) [pdf, html, other]
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Title: Detectors for local discrimination of sets of generalized Bell statesComments: 10 pages, 2 figures,7 tablesJournal-ref: Physical Review A 111, 042408 (2025)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
A fundamental problem in quantum information processing is the discrimination among a set of orthogonal quantum states of a composite system under local operations and classical communication (LOCC). Corresponding to the LOCC indistinguishable sets of four ququad-ququad orthogonal maximally entangled states (MESs) constructed by Yu et al. [Phys. Rev. Lett. 109, 020506 (2012)], the maximum commutative sets (MCSs) were introduced as detectors for the local distinguishability of the set of generalized Bell states (GBSs), for which the detectors are sufficient to determine the LOCC distinguishability. In this work, we show how to determine all the detectors for a given GBS set. We construct also several 4-GBS sets without detectors, most of which are one-way LOCC indistinguishable and only one is one-way LOCC distinguishable, indicating that the detectors are not necessary for LOCC distinguishability. Furthermore, we show that for 4-GBS sets in quantum system $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$, the detectors are almost necessary for one-way LOCC distinguishability, except for one set in the sense of local unitary equivalence. The problem of one-way LOCC discrimination of 4-GBS sets in $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$ is completely resolved.
- [9] arXiv:2504.11938 (cross-list from quant-ph) [pdf, html, other]
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Title: Exact noise and dissipation operators for quantum stochastic thermodynamicsSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker-Planck and Klein-Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator not only preserves the mathematical rigor of open quantum system dynamics but also allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for whom it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.
- [10] arXiv:2504.12081 (cross-list from hep-th) [pdf, html, other]
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Title: QCD$_2$ 't Hooft model: 2-flavour mesons spectrumComments: 47 pages and 5 figuresSubjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
We continue analytical study of the meson mass spectrum in the large-$N_c$ two-dimensional QCD, known as the 't Hooft model, by addressing the most general case of quarks with unequal masses. Based on our previous work, we develop non-perturbative methods to compute spectral sums and systematically derive large-$n$ WKB expansion of the spectrum. Furthermore, we examine the behavior of these results in various asymptotic regimes, including the chiral, heavy quark, and heavy-light limits, and establish a precise coincidence with known analytical and numerical results obtained through alternative approaches.
- [11] arXiv:2504.12263 (cross-list from quant-ph) [pdf, other]
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Title: A complete theory of the Clifford commutantComments: 84 pagesSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group -a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with $k$-fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of $k$, constrained by the number of qubits $n$. In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary $n$ and $k$. We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations -which generate the unitary group commutant- and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.
- [12] arXiv:2504.12280 (cross-list from physics.flu-dyn) [pdf, html, other]
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Title: On the range of validity of parabolic models for fluid flow through isotropic homogeneous porous mediaSubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph)
Lattice-Boltzmann methods are established mesoscopic numerical schemes for fluid flow, that recover the evolution of macroscopic quantities (viz., velocity and pressure fields) evolving under macroscopic target equations. The approximated target equations for fluid flows are typically parabolic and include a (weak) compressibility term. A number of Lattice-Boltzmann models targeting, or making use of, flow through porous media in the representative elementary volume, have been successfully developed. However, apart from two exceptions, the target equations are not reported, or the assumptions for and approximations of these equations are not fully clarified. Within this work, the underlying assumption underpinning parabolic equations for porous flow in the representative elementary volume, are discussed, clarified and listed. It is shown that the commonly-adopted assumption of negligible hydraulic dispersion is not justifiable by clear argument - and in fact, that by not adopting it, one can provide a qualitative and quantitative expression for the effective viscosity in the Brinkman correction of Darcy law. Finally, it is shown that, under certain conditions, it is possible to interpret porous models as Euler-Euler multiphase models wherein one phase is the solid matrix.
Cross submissions (showing 9 of 9 entries)
- [13] arXiv:2403.08965 (replaced) [pdf, html, other]
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Title: Deep Learning Based Dynamics Identification and Linearization of Orbital Problems using Koopman TheorySubjects: Mathematical Physics (math-ph); Earth and Planetary Astrophysics (astro-ph.EP); Machine Learning (cs.LG); Space Physics (physics.space-ph)
The study of the Two-Body and Circular Restricted Three-Body Problems in the field of aerospace engineering and sciences is deeply important because they help describe the motion of both celestial and artificial satellites. With the growing demand for satellites and satellite formation flying, fast and efficient control of these systems is becoming ever more important. Global linearization of these systems allows engineers to employ methods of control in order to achieve these desired results. We propose a data-driven framework for simultaneous system identification and global linearization of the Circular, Elliptical and Perturbed Two-Body Problem as well as the Circular Restricted Three-Body Problem around the L1 Lagrange point via deep learning-based Koopman Theory, i.e., a framework that can identify the underlying dynamics and globally linearize it into a linear time-invariant (LTI) system. The linear Koopman operator is discovered through purely data-driven training of a Deep Neural Network with a custom architecture. This paper displays the ability of the Koopman operator to generalize to various other Two-Body systems without the need for retraining. We also demonstrate the capability of the same architecture to be utilized to accurately learn a Koopman operator that approximates the Circular Restricted Three-Body Problem.
- [14] arXiv:2410.21173 (replaced) [pdf, html, other]
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Title: Nonlinear subwavelength resonances in three dimensionsSubjects: Mathematical Physics (math-ph)
In this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators with Kerr-type nonlinearities. Our discrete formulation is valid in both weak and strong nonlinear regimes. Compared to the linear formulation, it characterizes the extra eigenmodes induced by the non-linearlity that have recently been experimentally observed.
- [15] arXiv:2410.22569 (replaced) [pdf, html, other]
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Title: Enhanced binding for a quantum particle coupled to scalar quantized fieldComments: 26 pagesSubjects: Mathematical Physics (math-ph); Probability (math.PR)
Enhanced binding of a quantum particle coupled to a quantized field means that the Hamiltonian of the particle alone does not have a bound state, while the particle-field Hamiltonian does. For the Pauli--Fierz model, this is usually shown via the binding condition, which works less well in the case of a linear coupling to a scalar field. In particular, the case of a single particle linearly coupled to a scalar field has been open so far. Using a method relying on functional integrals and the Gaussian correlation inequality, we obtain enhanced binding for this case. From a statistical mechanics point of view, our result describes a localization phase transition (in the strength of the pair potential) for a Brownian motion subject to an external and an attractive pair potential.
- [16] arXiv:2504.10741 (replaced) [pdf, html, other]
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Title: $q$-Heisenberg Algebra in $\otimes^{2}-$Tensor SpaceSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
In this paper, we introduce the $q$-Heisenberg algebra in the tensor product space $\otimes^2$. We establish its algebraic properties and provide applications to the theory of non-monogenic functions. Our results extend known constructions in $q$-deformed algebras and offer new insights into functional analysis in non-commutative settings.
- [17] arXiv:2308.08706 (replaced) [pdf, html, other]
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Title: Bures geodesics and quantum metrologyComments: 23 pages, 3 figuresSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
We study the geodesics on the manifold of mixed quantum states for the Bures metric. It is shown that these geodesics correspond to physical non-Markovian evolutions of the system coupled to an ancilla. Furthermore, we argue that geodesics lead to optimal precision in single-parameter estimation in quantum metrology. More precisely, if the unknown parameter $x$ is a phase shift proportional to the time parametrizing the geodesic, the estimation error obtained by processing the data of measurements on the system is equal to the smallest error that can be achieved from joint detections on the system and ancilla, meaning that there is no information loss on this parameter in the ancilla. This error can saturate the Heisenberg bound. Reciprocally, assuming that the system-ancilla output and input states are related by a unitary $e^{-i x H}$ with $H$ a $x$-independent Hamiltonian, we show that if the error obtained from measurements on the system is equal to the minimal error obtained from joint measurements on the system and ancilla then the system evolution is given by a geodesic. In such a case, the measurement on the system bringing most information on $x$ is $x$-independent and can be determined in terms of the intersections of the geodesic with the boundary of quantum states. These results show that geodesic evolutions are of interest for high-precision detections in systems coupled to an ancilla in the absence of measurements on the ancilla.
- [18] arXiv:2309.02074 (replaced) [pdf, html, other]
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Title: Approximate recoverability and the quantum data processing inequalityComments: 25 pages, 0 figuresSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
In this paper, we discuss the quantum data processing inequality and its refinements that are physically meaningful in the context of approximate recoverability. An important conjecture regarding this due to Seshadreesan et. al. in J. Phys. A: Math. Theor. 48 (2015) is disproved. We prove some inequalities capturing universal approximate recoverability with the Petz recovery map for the sandwiched quasi and Rényi relative entropies for the parameter $t=2$. We also obtain convexity theorems on some parametrized versions of the relative entropy and fidelity, which can be of independent interest.
- [19] arXiv:2310.14937 (replaced) [pdf, html, other]
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Title: On the generalized Friedrichs-Lee model with multiple discrete and continuous statesComments: 29 pages, 3 figures, accepted verion for publication in Chinese Physics CSubjects: High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph); Nuclear Theory (nucl-th); Quantum Physics (quant-ph)
In this study, we present several improvements of the non-relativistic Friedrichs-Lee model with multiple discrete and continuous states and still retain its solvability. Our findings establish a solid theoretical basis for the exploration of resonance phenomena in scenarios involving multiple interfering states across various channels. The scattering amplitudes associated with the continuum states naturally adhere to coupled-channel unitarity, rendering this framework particularly valuable for investigating hadronic resonant states appearing in multiple coupled channels. Moreover, this generalized framework exhibits a wide-range applicability, enabling investigations into resonance phenomena across diverse physical domains, including hadron physics, nuclear physics, optics, and cold atom physics, among others.
- [20] arXiv:2402.09981 (replaced) [pdf, html, other]
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Title: Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle pointsComments: 14 pages, 7 figures, v3: we corrected minor typos in equations in the introduction and Section. IVJournal-ref: Phys.Rev.D 110 (2024) 8, 083508Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We revisit the path-integral approach to the wave function of the Universe by utilizing Lefschetz thimble analyses and resurgence theory. The traditional Euclidean path-integral of gravity has the notorious ambiguity of the direction of Wick rotation. In contrast, the Lorentzian method can be formulated concretely with the Picard-Lefschetz theory. Yet, a challenge remains: the physical parameter space lies on a Stokes line, meaning that the Lefschetz-thimble structure is still unclear. Through complex deformations, we resolve this issue by uniquely identifying the thimble structure. This leads to the tunneling wave function, as opposed to the no-boundary wave function, offering a more rigorous proof of the previous results. Further exploring the parameter space, we discover rich structures: the ambiguity of the Borel resummation of perturbative series around the tunneling saddle points is exactly canceled by the ambiguity of the contributions from no-boundary saddle points. This indicates that resurgence also works in quantum cosmology, particularly in the minisuperspace model.
- [21] arXiv:2402.17626 (replaced) [pdf, html, other]
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Title: An algebraic approach to gravitational quantum mechanicsComments: 19 pages, 2 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Most approaches towards a quantum theory of gravitation indicate the existence of a minimal length scale of the order of the Planck length. Quantum mechanical models incorporating such an intrinsic length scale call for a deformation of Heisenberg's algebra resulting in a generalized uncertainty principle and constitute what is called gravitational quantum mechanics. Utilizing the position representation of this deformed algebra, we study various models of gravitational quantum mechanics. The free time evolution of a Gaussian wave packet is investigated as well as the spectral properties of a particle bound by an external attractive potential. Here the cases of a box with infinite walls and an attractive potential well of finite depth are considered.
- [22] arXiv:2404.14997 (replaced) [pdf, html, other]
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Title: Mining higher-order triadic interactionsMarta Niedostatek, Anthony Baptista, Jun Yamamoto, Ben MacArthur, Jurgen Kurths, Ruben Sanchez Garcia, Ginestra BianconiSubjects: Adaptation and Self-Organizing Systems (nlin.AO); Statistical Mechanics (cond-mat.stat-mech); Social and Information Networks (cs.SI); Mathematical Physics (math-ph); Physics and Society (physics.soc-ph)
Complex systems often involve higher-order interactions which require us to go beyond their description in terms of pairwise networks. Triadic interactions are a fundamental type of higher-order interaction that occurs when one node regulates the interaction between two other nodes. Triadic interactions are found in a large variety of biological systems, from neuron-glia interactions to gene-regulation and ecosystems. However, triadic interactions have so far been mostly neglected. In this article, we propose a theoretical model that demonstrates that triadic interactions can modulate the mutual information between the dynamical state of two linked nodes. Leveraging this result, we propose the Triadic Interaction Mining (TRIM) algorithm to mine triadic interactions from node metadata, and we apply this framework to gene expression data, finding new candidates for triadic interactions relevant for Acute Myeloid Leukemia. Our work reveals important aspects of higher-order triadic interactions that are often ignored, yet can transform our understanding of complex systems and be applied to a large variety of systems ranging from biology to the climate.
- [23] arXiv:2405.04076 (replaced) [pdf, html, other]
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Title: 2d Sinh-Gordon model on the infinite cylinderComments: 37 pages, fixed typos + added refsSubjects: Probability (math.PR); Mathematical Physics (math-ph)
For $R>0$, we give a rigorous probabilistic construction on the cylinder $\mathbb{R} \times (\mathbb{R}/(2\pi R\mathbb{Z}))$ of the (massless) Sinh-Gordon model. In particular we define the $n$-point correlation functions of the model and show that these exhibit a scaling relation with respect to $R$. The construction, which relies on the massless Gaussian Free Field, is based on the spectral analysis of a quantum operator associated to the model. Using the theory of Gaussian multiplicative chaos, we prove that this operator has discrete spectrum and a strictly positive ground state.
- [24] arXiv:2407.21653 (replaced) [pdf, html, other]
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Title: Grothendieck Shenanigans: Permutons from pipe dreams via integrable probabilityComments: 50 pages; 16 figures; 2 tables. Including C code for random permutations and a CSV list of optimal layered permutations for beta=1 Grothendieck polynomials as ancillary files. v2: minor typos; added asymptotic number of inversions for arbitrary p; fixed proof of Grothendieck specialisations for layered permutations (statement does not change); fixed C code and updated graphs for 0<q<1Subjects: Probability (math.PR); Mathematical Physics (math-ph); Combinatorics (math.CO)
We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $\beta=1$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $n$ of the permutation grows to infinity. The fluctuations are of order $n^{\frac13}$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class.
We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $\beta=1$ Grothendieck polynomials, and provide bounds for general $\beta$. - [25] arXiv:2408.00535 (replaced) [pdf, html, other]
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Title: An explicit formula for free multiplicative Brownian motions via spherical functionsComments: Minor corrections and some comments addedSubjects: Probability (math.PR); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
After some normalization, the logarithms of the ordered singular values of Brownian motions on $GL(N,\mathbb F)$ with $\mathbb F=\mathbb R, \mathbb C$ form Weyl-group invariant Heckman-Opdam processes on $\mathbb R^N$ of type $A_{N-1}$. We use classical elementary formulas for the spherical functions of $GL(N,\mathbb C)/SU(N)$ and the associated Euclidean spaces $H(N,\mathbb C)$ of Hermitian matrices, and show that in the $GL(N,\mathbb C)$-case, these processes can be also interpreted as ordered eigenvalues of Brownian motions on $H(N,\mathbb C)$ with particular drifts. This leads to an explicit description for the free limits for the associated empirical processes for $N\to\infty$ where these limits are independent from the parameter $k$ of the Heckman-Opdam processes. In particular we get new formulas for the distributions of the free multiplicative Browniam motion of Biane. We also show how this approach works for the root systems $B_N, C_N, D_N$.
- [26] arXiv:2409.11324 (replaced) [pdf, html, other]
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Title: Meson mass spectrum in QCD$_2$ 't Hooft's modelComments: 29 pages and 1 figure, v3: typos correctedSubjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
We study the spectrum of meson masses in large $N_c$ QCD$_2$ governed by celebrated 't Hooft's integral equation. We generalize analytical methods proposed by Fateev, Lukyanov and Zamolodchikov to the case of arbitrary, but equal quark masses $m_1=m_2.$ Our results include analytical expressions for spectral sums and systematic large-$n$ expansion. We also study the spectral sums in the chiral limit and the heavy quark limit and find a complete agreement with known results.
- [27] arXiv:2411.04183 (replaced) [pdf, html, other]
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Title: Superadditivity in large $N$ field theories and performance of quantum tasksComments: 68 pages, 18 figuresSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
Field theories exhibit dramatic changes in the structure of their operator algebras in the limit where the number of local degrees of freedom ($N$) becomes infinite. An important example of this is that the algebras associated to local subregions may not be additively generated in the limit. We investigate examples and explore the consequences of this ``superadditivity'' phenomenon in large $N$ field theories and holographic systems. In holographic examples we find cases in which superadditive algebras can probe the black hole interior, while the additive algebra cannot. We also discuss how superaddivity explains the sucess of quantum error correction models of holography. Finally we demonstrate how superadditivity is intimately related to the ability of holographic field theories to perform quantum tasks that would naievely be impossible. We argue that the connected wedge theorems (CWTs) of May, Penington, Sorce, and Yoshida, which characterize holographic protocols for quantum tasks, can be re-phrased in terms of superadditive algebras and use this re-phrasing to conjecture a generalization of the CWTs that is an equivalence statement.
- [28] arXiv:2411.08311 (replaced) [pdf, html, other]
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Title: Martingale properties of entropy production and a generalized work theorem with decoupled forward and backward processesSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)
By decoupling forward and backward stochastic trajectories, we construct a family of martingales and work theorems for both overdamped and underdamped Langevin dynamics. Our results are made possible by an alternative derivation of work theorems that uses tools from stochastic calculus instead of path-integration. We further strengthen the equality in work theorems by evaluating expectations conditioned on an arbitrary initial state value. These generalizations extend the applicability of work theorems and offer new interpretations of entropy production in stochastic systems. Lastly, we discuss the violation of work theorems in far-from-equilibrium systems.
- [29] arXiv:2412.05639 (replaced) [pdf, html, other]
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Title: On the Equivalence of Equilibrium and Freezing States in Dynamical SystemsSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
This paper is concerned with freezing phase transitions in general dynamical systems. A freezing phase transition is one in which, for a given potential $\phi$, there exists some inverse temperature $\beta_0 > 0$ such that for all $\alpha, \beta > \beta_0$, the collection of equilibrium states for $\alpha \phi$ and $\beta \phi$ coincide. In this sense, below the temperature $1 / \beta_0$, the system "freezes" on a fixed collection of equilibrium states.
We show that for a given invariant measure $\mu$, it is no more restrictive that $\mu$ is the freezing state for some potential than it is for $\mu$ to be the equilibrium state for some potential. In fact, our main result applies to any collection of equilibrium states with the same entropy. In the case where the entropy map $h$ is upper semi-continuous, we show any ergodic measure $\mu$ can be obtained as a freezing state for some potential.
In this upper semi-continuous setting, we additionally show that the collection of potentials that freeze at a single state is dense in the space of all potentials. However, in the $\Z$ action setting where the dynamical system satisfies specification, the collection of potentials that do not freeze contains a dense $G_\delta$. - [30] arXiv:2501.09413 (replaced) [pdf, html, other]
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Title: Quantum algorithm for the gradient of a logarithm-determinantComments: 20 pages, 3 figures, 2 circuit diagrams, 1 tableSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
The logarithm-determinant is a common quantity in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories, as well as the inverses of matrices. A multi-variable version of the quantum gradient algorithm is developed here to evaluate the derivative of the logarithm-determinant. From this, the inverse of a sparse-rank input operator may be determined efficiently. Measuring an expectation value of the quantum state--instead of all $N^2$ elements of the input operator--can be accomplished in $O(k\sigma)$ time in the idealized case for $k$ relevant eigenvectors of the input matrix. A factor $\sigma=\frac1{\varepsilon^3}$ or $-\frac1{\varepsilon^2}\log_2\varepsilon$ depends on the version of the quantum Fourier transform used for a precision $\varepsilon$. Practical implementation of the required operator will likely need $\log_2N$ overhead, giving an overall complexity of $O(k\sigma\log_2 N)$. The method applies widely and converges super-linearly in $k$ when the condition number is high. The best classical method we are aware of scales as $N$.
Given the same resource assumptions as other algorithms, such that an equal superposition of eigenvectors is available efficiently, the algorithm is evaluated in the practical case as $O(\sigma\log_2 N)$. The output is given in $O(1)$ queries of oracle, which is given explicitly here and only relies on time-evolution operators that can be implemented with arbitrarily small error. The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines. We discuss how this algorithm can be used for kernel-based quantum machine-learning. - [31] arXiv:2503.24060 (replaced) [pdf, html, other]
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Title: Quantization of Lie-Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix modelComments: v2: references and comments added, typos corrected, minor changes to discussion ; 45 pages, 1 figure, v3: No change in content, Fixing display errors in the abstract textSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
A quantization of Lie-Poisson algebras is studied. Classical solutions of the mass-deformed IKKT matrix model can be constructed from semisimple Lie algebras whose dimension matches the number of matrices in the model. We consider the geometry described by the classical solutions of the Lie algebras in the limit where the mass vanishes and the matrix size tends to infinity. Lie-Poisson varieties are regarded as such geometric this http URL provide a quantization called ``weak matrix regularization''of Lie-Poisson algebras (linear Poisson algebras) on the algebraic varieties defined by their Casimir polynomials. Casimir polynomials correspond with Casimir operators of the Lie algebra by the quantization. This quantization is a generalization of the method for constructing the fuzzy sphere. In order to define the weak matrix regularization of the quotient space by the ideal generated by the Casimir polynomials, we take a fixed reduced Gröbner basis of the ideal. The Gröbner basis determines remainders of polynomials. The operation of replacing this remainders with representation matrices of a Lie algebra roughly corresponds to a weak matrix regularization. As concrete examples, we construct weak matrix regularization for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$. In the case of $\mathfrak{su}(3)$, we not only construct weak matrix regularization for the quadratic Casimir polynomial, but also construct weak matrix regularization for the cubic Casimir polynomial.
- [32] arXiv:2504.00269 (replaced) [pdf, other]
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Title: Existence of Full Replica Symmetry Breaking for the Sherrington-Kirkpatrick Model at Low TemperatureComments: 19 pages, 2 figures. Results improved in Theorem 1 and Corollary 2Subjects: Probability (math.PR); Mathematical Physics (math-ph)
We prove the existence of full replica symmetry breaking (FRSB) for the Sherrington-Kirkpatrick (SK) model at low temperature. More specifically, we prove that slightly beyond the critical temperature, the Parisi measure for the SK model is supported on an interval starting at the origin and only has one jump discontinuity at the right endpoint.
