# Mathematical Physics

## New submissions

[ total of 20 entries: 1-20 ]
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### New submissions for Thu, 23 Nov 17

[1]
Title: Blocks in the Asymmetric Simple Exclusion Process: Asymptotics
Subjects: Mathematical Physics (math-ph)

In earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time $t$ a particle is at site $x$ and is the beginning of a block of $L$ consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\to\infty$) that a particle is the beginning of an $L$-block, given that it is at site $x$ at time $t$.

[2]
Title: Soap film spanning an elastic link
Subjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

We study the equilibrium problem of a system consisting by several Kirchhoff rods linked in an arbitrary way and tied by a soap film, using techniques of the Calculus of Variations. We prove the existence of a solution with minimum energy, which may be quite irregular, and perform experiments confirming the kind of surface predicted by the model.

[3]
Title: The shape of the emerging condensate in effective models of condensation
Subjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the build-up of the condensate occurs on a spatial scale of 1/t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose-Einstein condensation.

[4]
Title: Holomorphic solutions of the susy $σ$-model and gauge invariance
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)

For the first time we develop the gauge invariance of the supersymmetric grassmannian sigma model $G(M,N)$. It is richer then its purely bosonic submodel and we show how to use it in order to reduce some constant curvature holomorphic solutions of the model into simpler expressions.

### Cross-lists for Thu, 23 Nov 17

[5]  arXiv:1711.08031 (cross-list from hep-th) [pdf, ps, other]
Title: Singular vector structure of quantum curves
Comments: 33 pages; proceedings of the 2016 AMS von Neumann Symposium
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.

[6]  arXiv:1711.08039 (cross-list from cs.CC) [pdf, other]
Title: Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory
Subjects: Computational Complexity (cs.CC); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Quantum Physics (quant-ph)

Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently.
In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem.
This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling.
Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems.
Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT).

[7]  arXiv:1711.08169 (cross-list from hep-th) [pdf, other]
Title: Notes on $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ representations
Authors: Alexei Kitaev
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)

These notes describe representations of the universal cover of $\mathrm{SL}(2,\mathbb{R})$ with a view toward applications in physics. Spinors on the hyperbolic plane and the two-dimensional anti-de Sitter space are also discussed.

[8]  arXiv:1711.08280 (cross-list from hep-th) [pdf, ps, other]
Title: AdS4 backgrounds with N>16 supersymmetries in 10 and 11 dimensions
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Differential Geometry (math.DG)

We explore all warped $AdS_4\times_w M^{D-4}$ backgrounds with the most general allowed fluxes that preserve more than 16 supersymmetries in $D=10$- and $11$-dimensional supergravities. After imposing the assumption that either the internal space $M^{D-4}$ is compact without boundary or the isometry algebra of the background decomposes into that of AdS$_4$ and that of $M^{D-4}$, we find that there are no such backgrounds in IIB supergravity. Similarly in IIA supergravity, there is a unique such background with 24 supersymmetries locally isometric to $AdS_4\times \mathbb{CP}^3$, and in $D=11$ supergravity all such backgrounds are locally isometric to the maximally supersymmetric $AdS_4\times S^7$ solution.

[9]  arXiv:1711.08304 (cross-list from math.FA) [pdf, ps, other]
Title: Boundary representation of Dirichlet forms on discrete spaces
Subjects: Functional Analysis (math.FA); Mathematical Physics (math-ph); Spectral Theory (math.SP)

We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.

[10]  arXiv:1711.08350 (cross-list from math.AP) [pdf, ps, other]
Title: Empirical Measures and Quantum Mechanics: Application to the Mean-Field Limit
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)

In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of $N$-particle systems. We establish an equation governing the evolution of our quantum analogue of the $N$-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Our main application of this new object to the mean-field limit of the $N$-particle Schr\"odinger equation is an $O(1/\sqrt{N})$ convergence rate in some dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\hbar\in(0,1]$ (where $\hbar$ is the Planck constant) provided that $V$ and $(-\Delta)^{3+d/2}V$ have integrable Fourier transforms.

[11]  arXiv:1711.08381 (cross-list from math.RA) [pdf, ps, other]
Title: Symplectic, product and complex structures on 3-Lie algebras
Subjects: Rings and Algebras (math.RA); Mathematical Physics (math-ph); Differential Geometry (math.DG)

In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to $\huaO$-operators, Rota-Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-K\"{a}hler structure and a pseudo-K\"{a}hler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.

[12]  arXiv:1711.08419 (cross-list from nlin.SI) [pdf, other]
Title: Nonlocal reductions of the Ablowitz-Ladik equation
Comments: 20 pages, 1 (png) figure, LaTeX
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Pattern Formation and Solitons (nlin.PS)

The purpose of the present paper is to develop the inverse scattering transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known as Ablowitz-Ladik equation) with PT-symmetry. This includes: the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data and the fundamental analytic solutions. In addition, the paper studies the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann-Hilbert problem, the 1- and 2-soliton solutions for the nonlocal Ablowitz-Ladik equation are derived. Finally, the completeness relation for the associated Jost solutions is proved. Based on this, the expansion formula over the complete set of Jost solutions is derived. This will allow one to interpret the inverse scattering transform as a generalised Fourier transform.

### Replacements for Thu, 23 Nov 17

[13]  arXiv:1407.4327 (replaced) [pdf, ps, other]
Title: Second-order integrals for systems in $E_2$ involving spin
Authors: Ismet Yurdusen
Journal-ref: Adv. Math. Phys. (2015) 1-7
Subjects: Mathematical Physics (math-ph)
[14]  arXiv:1701.00116 (replaced) [pdf, other]
Title: A rigourous demonstration of the validity of Boltzmann's scenario for the spatial homogenization of a freely expanding gas and the equilibration of the Kac ring
Journal-ref: J Stat Phys (2017) 168: 772
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)
[15]  arXiv:1705.03246 (replaced) [pdf, ps, other]
Title: Two-dimensional position-dependent mass Lagrangians; Superintegrability and exact solvability
Authors: Omar Mustafa
Subjects: Mathematical Physics (math-ph)
[16]  arXiv:1706.09543 (replaced) [pdf, ps, other]
Title: Absence of replica symmetry breaking in the transverse and longitudinal random field Ising model
Authors: C. Itoi
Subjects: Mathematical Physics (math-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Quantum Physics (quant-ph)
[17]  arXiv:1708.09460 (replaced) [pdf, ps, other]
Title: The Hammersley-Welsh bound for self-avoiding walk revisited
Authors: Tom Hutchcroft
Comments: 9 pages. V2: fixed typo in abstract. V3: Errors corrected plus some other minor revisions. To appear in ECP
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Combinatorics (math.CO)
[18]  arXiv:1711.02063 (replaced) [pdf, other]
Title: Cluster integrable systems, q-Painleve equations and their quantization