Subhajit Goswami, Pierre-François Rodriguez, Yuriy Shulzhenko

We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal exponential order for all $u$ in dimension three and all $u \neq u_\ast$ in higher dimensions, where $u_*$ refers to the critical parameter of the model, and they match the upper bounds derived in the article arXiv:2503.14497. In dimension three, our results further imply that the truncated two-point function grows at large distances $x$ at a rate that depends on $x$ only through its Euclidean norm, which offers a glimpse of the expected (Euclidean) invariance of the scaling limit at criticality. The rate function is atypical, it incurs a logarithmic correction and comes with an explicit pre-factor that converges to $0$ as the parameter $u$ approaches the critical point $u_*$ from either side. A particular challenge stems from the combined effects of lack of monotonicity due to the truncation in the super-critical phase, and the precise (rotationally invariant) controls we seek, that measure the effects of a certain "harmonic humpback" function. Among others, their derivation relies on rather fine estimates for hitting probabilities of the random walk in arbitrary direction $e$, which witness this invariance at the discrete level, and preclude straightforward applications of projection arguments. Read complete

Benedetta Bertoli, Grigorios A. Pavliotis, Niccolò Zagli

In this paper, we study weakly interacting diffusion processes on random graphs. Our main focus is on the properties of the mean-field limit and, in particular, on the nonuniqueness of stationary states. By extending classical bifurcation analysis to include multichromatic interaction potentials and random graph structures, we explicitly identify bifurcation points and relate them to the eigenvalues of the graphon integral operator. Furthermore, we characterize the resulting McKean-Vlasov PDE as a gradient flow with respect to a suitable metric. We combine these theoretical results with the spectral analysis of the linearized McKean-Vlasov operator and extensive numerical simulations to gain insight into the stability and long-term behaviour of stationary solutions. In addition, we provide strong evidence that (minus) the interaction energy of the interacting particle system serves as a natural order parameter. In particular, beyond the transition point and for multichromatic interactions, we observe an energy cascade that is strongly linked to the dynamical metastability of the system. Read complete

Zachary Mullaghy

We present a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional. By modeling interior point distributions in a convex domain as repelling charges balanced against a continuous boundary charge, we show that the resulting equilibrium configurations converge to CVT structures. We prove this by showing that CVTs minimize both the classical centroidal energy and the electrostatic potential, establishing a connection between geometric quantization and potential theory. Finally, we introduce a thermodynamic annealing scheme for global CVT optimization, rooted in Boltzmann statistics and random walk dynamics. By introducing a scheme for varying time steps (faster or slower cooling) we show that the set of minima of the centroid energy functional (and therefore the electrostatic potential) can be recovered. By recovering a set of generator locations corresponding to each minimum we can create a lattice continuation that allows for a customizable framework for individual minimum seeking. Read complete

Kristan Temme, Pawel Wocjan

We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a completely positive and trace preserving map. This quantum map has a unique fixed point, which corresponds to the quantum sample (qsample) of the classical Markov chain's stationary distribution. We show that the convergence time of the quantum map is directly related to the coupling time of the Markov chain coupling. Read complete

Chen Sun

The Landau-Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau-Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and mathematically simple. Our work provides new insight into the origin of the exponential form of the Landau-Zener transition probability. Read complete

Raphaël Côte, Clémentine Courtès, Guillaume Ferrière, Ludovic Godard-Cadillac, Yannick Privat

We study from a mathematical point of view the nanoparticle model of a magnetic colloid, presented by G. Klughertz. Our objective is to obtain properties of stable stationary structures that arise in the long-time limit for the magnetic nanoparticles dynamics following this model. In this article, we present a detailed study of two specific structures using techniques from the calculus of variations. The first, called the spear, consists of a chain of aligned particles interacting via a Lennard-Jones potential. We establish existence and uniqueness results, derive bounds on the distances between neighboring particles, and provide a sharp asymptotic description as the number of particles tends to infinity. The second structure, the ring, features particles uniformly distributed along a circle. We prove its existence and uniqueness and derive an explicit formula for its radius. Read complete

V. M. Jiménez, M. De León, M. Epstein

In this paper we present a geometrical framework to study the uniformity of a composite material by means of double groupoid theory. The notions of vertical and horizontal uniformity are introduced, as well as other weaker ones that allows us to study other possible notions of more general uniformity. Read complete

Sourav Chatterjee

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this "wrong sign" requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$-point correlation function when the parameters satisfy the so-called "charge neutrality condition". Expressions are derived also for the $k$-point correlation functions for all $k\ge 3$, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero. Read complete

Matteo Capoferri, Beatrice Costeri, Claudio Dappiaggi

A new approach to the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed manifolds was proposed in a recent series of papers by the first author and collaborator. They showed that information on spectral asymmetry can be encoded within and recovered from a negative order pseudodifferential operator -- the asymmetry operator -- constructed from appropriately defined pseudodifferential (spectral) projections. In this manuscript we apply these techniques to the study of the massless Dirac operator; in particular, we compute the principal symbol of the asymmetry operator, accounting for the underlying gauge invariance. Read complete

Pavel Etingof, Ivan Motorin, Alexander Varchenko, Isaac Zhu

We consider the Knizhnik-Zamolodchikov equations in Deligne Categories in the context of $(\mathfrak{gl}_m,\mathfrak{gl}_{n})$ and $(\mathfrak{so}_m,\mathfrak{so}_{2n})$ dualities. We derive integral formulas for the solutions in the first case and compute monodromy in both cases. Read complete

Kohtaro Kato, Tomotaka Kuwahara

Conditional mutual information (CMI) has recently attracted significant attention as a key quantity for characterizing quantum correlations in many-body systems. While it is conjectured that CMI decays rapidly in finite-temperature Gibbs states, a complete and general proof remains elusive. Previous work addressed this problem in the high-temperature regime using cluster expansion techniques [T. Kuwahara, K. Kato, F.G.S.L. Brand\~ao, Phys. Rev. Lett. 124, 220601 (2020)]; however, flaws in the proof have been pointed out, and the method does not provide a uniformly convergent expansion at arbitrarily high temperatures. In this work, we demonstrate that the cluster expansion approach indeed fails to converge absolutely, even at any high-temperatures. To overcome this limitation, we propose a new approach to proving the spatial decay of CMI. Our method leverages the connection between CMI and quantum recovery maps, specifically utilizing the Fawzi-Renner theorem. We show that such recovery maps can be realized through dissipative dynamics, and by analyzing the locality properties of these dynamics, we establish the exponential decay of CMI in high-temperature regimes. As a technical contribution, we also present a new result on the perturbative stability of quasi-local Liouvillian dynamics. Our results indicate that, contrary to common intuition, high-temperature Gibbs states can exhibit nontrivial mathematical structure, particularly when multipartite correlations such as CMI are considered. Read complete

Chi-Fang Chen, Cambyse Rouzé

The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree, the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on $D$-dimensional lattices can be prepared by a quantum circuit of depth $e^{O(\log^D(n/\epsilon))}$, which can be further reduced assuming certain local gap condition. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes. Read complete

Sasa Kresic-Juric, Concepcion Muriel, Adrian Ruiz

In this paper, we investigate solvable structures associated to Hamiltonian equations. For a completely integrable Hamiltonian system with $n$ degrees of freedom, we construct a canonical solvable structure consisting of $2n$ Hamiltonian vector fields. We derive explicit expressions for the corresponding Pfaffian forms, whose integration provides solutions to the Hamiltonian equations. We show that the upper $n$ forms give the action varibles, while the lower $n$ forms yield the angle variables of the system. This offers a novel interpretation of the Arnold--Liouville theorem in terms of solvable structures. We ilustrate the theory by deriving explicit solutions and action--angle variables for $n$ harmonic oscillators and the Calogero--Moser system. Read complete

Théophile Dolmaire, Nicola Miele, Alessia Nota

In this article we consider the linear inelastic Boltzmann equation in presence of a uniform and fixed gravity field, in the case of Maxwell molecules. We first obtain a well-posedness result in the space of finite, non-negative Radon measures. In addition, we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the external field. We further show that this stationary solution is unique in the class of the finite, non-negative Radon measures with finite first order moment, and that all the solutions in this class converge towards the stationary solution in the weak topology of the measures. Read complete

Francesco Toppan

I point out that a possible minimal setting to realize Kitaev's proposal of a Topological Quantum Computation which offers topological protection from decoherence could in principle be realized by braided Majorana qubits. Majorana qubits and their braiding were introduced in Nucl. Phys. B 980, 115834 (2022) and further analyzed in J. Phys. A: Math. Theor. 57, 435203 (2024). Braided Majorana qubits implement a Gentile-type parastatistics with at most $s-1$ excited states accommodated in a multiparticle sector (the integer $s=2,3,4,\ldots$ labels quantum group reps at roots of unity). It is argued that braided Majorana qubits could play, for topological quantum computers, the same role as standard bits for ordinary computers and as qubits for "ordinary" quantum computers. Read complete

Alexandr Buryak, Paolo Rossi

The Riemann hierarchy is the simplest example of rank one, ($1$+$1$)-dimensional integrable system of nonlinear evolutionary PDEs. It corresponds to the dispersionless limit of the Korteweg--de Vries hierarchy. In the language of formal variational calculus, we address the classification problem for deformations of the Riemann hierarchy satisfiying different extra requirements (general deformations, defomations as systems of conservation laws, Hamiltonian deformations, and tau-symmetric deformations), under the natural group of coordinate transformations preserving each of those requirements. We present several results linking previous conjectures of Dubrovin--Liu--Yang--Zhang (for the tau-symmetric case) and of Arsie--Lorenzoni--Moro (for systems of conservation laws) to the double ramification hierarchy construction of integrable hierarchies from partial CohFTs and F-CohFTs. We prove that, if the conjectures are true, DR hierarchies of rank one are universal objects in the space of deformations of the Riemann hierarchy. We also prove a weaker version of the DLYZ conjecture and that the ALM conjecture implies (the main part of) the DLYZ conjecture. Finally we characterize those rank one F-CohFTs which give rise to Hamiltonian deformations of the Riemann hierarchy. Read complete

A. Mironov, A. Morozov, A. Popolitov

General description of eigenfunctions of integrable Hamiltonians associated with the integer rays of Ding-Iohara-Miki (DIM) algebra, is provided by the theory of Chalykh Baker-Akhiezer functions (BAF) defined as solutions to a simply looking linear system. Solutions themselves are somewhat complicated, but much simpler than they could. It is because of simultaneous partial factorization of all the determinants, entering Cramer's rule. This is a conspiracy responsible for a relative simplicity of the Macdonald polynomials and of the Noumi-Shirashi functions, and it is further continued to all integer DIM rays. Still, factorization is only partial, moreover, there are different branches and abrupt jumps between them. We explain this feature of Cramer's rule in an example of a matrix that defines BAF and exhibits a non-analytical dependence on parameters. Moreover, the matrix is such that there is no natural expansion around non-degenerate approximations, which causes an unexpected complexity of formulas. Read complete

Ion Nechita, Sang-Jun Park

Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this work, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the \emph{tensor distribution} limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of freeness, which we term \emph{tensor freeness}. It can be defined via the vanishing of mixed \emph{tensor free cumulants}, allowing the joint tensor distribution of tensor free elements to be described in terms of that of individual elements. We present several applications of these results in the context of random matrices with a tensor product structure, such as partial transpositions of (local) unitarily invariant random matrices and tensor embeddings of random matrices. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables. Read complete

Markus Penz, Andre Laestadius

We study a generalization of Moreau-Yosida regularization that is adapted to the geometry of Banach spaces where the dual space is uniformly convex with modulus of convexity of power type. Important properties for regularized convex functions are given, in particular strong monotonicity of the subdifferential of their convex conjugate and H\"older-continuity of their gradient. Read complete

I. V. Anikin

An alternative method to invert the Radon transforms without the use of Courand-Hilbert's identities has been proposed and developed independently from the space dimension. For the universal representation of inverse Radon transform, we study the consequences of inhomogeneity of outset function without the restrictions on the angular Radon coordinates. We show that this inhomogeneity yields a natural evidence for the presence of the extra contributions in the case of the full angular region. In addition, if the outset function is well-localized in the space, we demonstrate that the corresponding boundary conditions and the angular restrictions should be applied for both the direct and inverse Radon transforms. Besides, we relate the angular restrictions on the Radon variable to the boundary exclusion of outset function and its Radon image. Read complete

Bernd Sturmfels, Máté L. Telek

Copositive matrices and copositive polynomials are objects from optimization. We connect these to the geometry of Feynman integrals in physics. The integral is guaranteed to converge if its kinematic parameters lie in the copositive cone. P\'olya's method makes this manifest. We study the copositive cone for the second Symanzik polynomial of any Feynman graph. Its algebraic boundary is described by Landau discriminants. Read complete

Martin Schneider, Hanchuan Wu, Maziar Veyskarami, Sorin Pop, Rainer Helmig

In fully-implicit two-phase pore-network models, developing a well-converged scheme remains a major challenge, primarily due to the discontinuities in the phase conductivities. This paper addresses these numerical issues by proposing a generalized flux function that establishes a continuous flux expression for two-phase flows by introducing an additional throat variable $\Theta$. Two approaches for expressing this additional throat variable are introduced: the first applies regularization strategies, while the second constructs an additional residual constraint equation. It is shown that this approach significantly improves accuracy and ensures the temporal convergence, as demonstrated through various numerical examples. Read complete

Jeremías Aguilera Damia, Riccardo Argurio, Antoine Bourget, Valdo Tatitscheff, Romain Vandepopeliere

Global variants of four-dimensional gauge theories are specified by their spectrum of genuine Wilson-'t Hooft line operators. The choice of global variant has significant consequences when spacetime is taken to be $\mathbb{R}^3 \times S^1$. We focus on $\mathcal N=1^*$ theories, which are closely connected to twisted elliptic Calogero-Moser systems. We establish, on general grounds, how this gauge-theoretic topological data manifests itself on the integrable system side by introducing a notion of global variants for complex many-body integrable systems associated with Lie algebras. Focusing on $\mathcal N=1^*$ theories of type $A$ and $B_2$, we elucidate the implications for the structure of gapped vacua, the emergent (generalized) symmetries realized in each vacuum, and the action of spontaneously broken modular invariance. Read complete

Manuela Feistl-Held, Peter Pickl

We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+\sigma}$ where $\sigma>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force. Read complete

Holger Dullin, Richard Montgomery

We show (Theorem 3) that the symplectic reduction of the spatial $n$-body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions. Each stratum is realized as coadjoint orbit in the dual of the Lie algebra of the linear symplectic group $Sp(2n-2)$. The planar stratum arises as the frontier upon taking the closure of the spatial stratum. We reduce by going to center-of-mass coordinates to reduce by translations and boosts and then performing symplectic reduction with respect to the orthogonal group $O(3)$. The theorem is a special case of a general theorem (Theorem 2) which holds for the $n$-body problem in any dimension $d$. This theorem follows largely from a ``Poisson reduction'' theorem, Theorem 1. We achieve our reduction theorems by combining the Howe dual pair perspective of reduction espoused by Lerman-Montgomery-Sjamaar with a normal form arising from a symplectic singular value decomposition due to Xu. We begin the paper by showing how Poisson reduction by the Galilean group rewrites Newton's equations for the $n$-body problem as a Lax pair. In section 6.4 we show that this Lax pair representation of the $n$-body equations is equivalent to the Albouy-Chenciner representation in terms of symmetric matrices. Read complete

Olga Boyko, Yuri Latushkin, Vyacheslav Pivovarchik

We consider a scattering problem generated by the Sturm-Liouville equation on a tree which consists of an equilateral compact subtree and a half-infinite lead attached to its root. We assume that the potential on the lead is identically zero while the potentials on the finite edges are real. We show how to find the shape of the tree using the S-function of the scattering problem and the eigenvalues of the operators associated with the compact tree. Read complete

Francisco Correa, Luis Inzunza, Olaf Lechtenfeld

It is long known that quantum Calogero models feature intertwining operators, which increase or decrease the coupling constant by an integer amount, for any fixed number of particles. We name these as ``horizontal'' and construct new ``vertical'' intertwiners, which \emph{change the number of interacting particles} for a fixed but integer value of the coupling constant. The emerging new structure of a grid of intertwiners exists only in the algebraically integrable situation (integer coupling) and allows one to obtain each Liouville charge from the free power sum in the particle momenta by iterated intertwining either horizontally or vertically. We present recursion formul\ae\ for the new intertwiners as a factorization problem for partial differential operators and prove their existence for small values of particle number and coupling. As a byproduct, a new basis of non-symmetric Liouville integrals appears, algebraically related to the standard symmetric one. Read complete

Krzysztof Szczygielski

We characterize covariant positive decomposable maps between unital C*-algebras in terms of a dilation theorem, which generalizes a seminal result by H. Scutaru from Rep. Math. Phys. 16 (1):79-87, 1979. As a case study, we provide a certain characterization of the operator sum representation of maps on $\mathbb{M}_n (\mathbb{C})$, covariant with respect to the maximal commutative subgroup of $\mathrm{U}(n)$. A connection to quantum dynamics is established by specifying sufficient and necessary conditions for covariance of D-divisible (decomposably divisible) quantum evolution families, recently introduced in J. Phys. A: Math. Theor. 56 (2023) 485202. Read complete

Daniel Coronel, Juan Rivera-Letelier

This article characterizes phase transitions in temperature within a specific space of H\"older continuous potentials, distinguished by their regularity and asymptotic behavior at zero. We also characterize the phase transitions in temperature that are robust within this space. Our results reveal a connection between phase transitions in temperature and ergodic optimization. Read complete

Max Emerick, Bassam Bamieh

The problem of incompressible fluid mixing arises in numerous engineering applications and has been well-studied over the years, yet many open questions remain. This paper aims to address the question "what do efficient flow fields for mixing look like, and how do they behave?" We approach this question by developing a framework which is inspired by the dynamic and geometric approach to optimal mass transport. Specifically, we formulate the fluid mixing problem as an optimal control problem where the dynamics are given by the continuity equation together with an incompressibility constraint. We show that within this framework, the set of reachable fluid configurations can formally be endowed with the structure of an infinite-dimensional Riemannian manifold, with a metric which is induced by the control effort, and that flow fields which are maximally efficient at mixing correspond to geodesics in this Riemannian space. Read complete

René Zander, Raphael Seidel, Li Xiaoyue, Marek Gluza

Double-bracket quantum imaginary-time evolution (DB-QITE) is a quantum algorithm which coherently implements steps in the Riemannian steepest-descent direction for the energy cost function. DB-QITE is derived from Brockett's double-bracket flow which exhibits saddle points where gradients vanish. In this work, we perform numerical simulations of DB-QITE and describe signatures of transitioning through the vicinity of such saddle points. We provide an explicit gate count analysis using quantum compilation programmed in Qrisp. Read complete

M. I. Belishev, S. A. Simonov

Let $\mathcal G$ be a Hilbert space and $\mathfrak B(\mathcal G)$ the algebra of bounded operators, $\mathcal H=L_2([0,\infty);\mathcal G)$. An operator-valued function $Q\in L_{\infty,\rm loc}\left([0,\infty);\mathfrak B(\mathcal G)\right)$ determines a multiplication operator in $\mathcal H$ by $(Qy)(x)=Q(x)y(x)$, $x\geqslant0$. We say that an operator $L_0$ in a Hilbert space is a Schr\"odinger type operator, if it is unitarily equivalent to $-d^2/dx^2+Q(x)$ on a relevant domain. The paper provides a characterization of a class of such operators. The characterization is given in terms of properties of an evolutionary dynamical system associated with $L_0$. It provides a way to construct a functional Schr\"odinger model of $L_0$. Read complete

V. M. Jiménez, M. De León

In this paper we discuss nonholonomic contact Lagrangian and Hamiltonian systems, that is, systems with a kind of dissipation that are also subject to nonholonomic constraints. We introduce the so-called contact Eden bracket that allows us to obtain the evolution of any observable. Finally, we present a particular vector subspace of observables where the dynamics remain unconstrained. Read complete

Alix Deleporte, Yohann Le Floch

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy levels. These results form a complex version of the Bohr-Sommerfeld quantization conditions; they hold under a hypothesis that the skew-adjoint part is small but can be of principal order with respect to the semiclassical parameter. To this end, we develop a calculus of Fourier Integral Operators and Lagrangian states associated with complex Lagrangians; these tools can be of independent interest. Read complete

Dmytro Sytnyk

In this work, we propose an exponentially convergent numerical method for the Caputo fractional propagator $S_\alpha(t)$ and the associated mild solution of the Cauchy problem with time-independent sectorial operator coefficient $A$ and Caputo fractional derivative of order $\alpha \in (0,2)$ in time. The proposed methods are constructed by generalizing the earlier developed approximation of $S_\alpha(t)$ with help of the subordination principle. Such technique permits us to eliminate the dependence of the main part of error estimate on $\alpha$, while preserving other computationally relevant properties of the original approximation: native support for multilevel parallelism, the ability to handle initial data with minimal spatial smoothness, and stable exponential convergence for all $t \in [0, T]$. Ultimately, the use of subordination leads to a significant improvement of the method's convergence behavior, particularly for small $\alpha < 0.5$, and opens up further opportunities for efficient data reuse. To validate theoretical results, we consider applications of the developed methods to the direct problem of solution approximation, as well as to the inverse problem of fractional order identification. Read complete

P. M. Akhmetiev, M. S. Dvornikov

We study an axion soliton, which weakly interacts with background matter and magnetic fields. A mirror-symmetric soliton, for which the magnetic flow is due to secondary magnetic helicity invariant, is described by the Iroshnikov-Kreichnan spectrum. For a large-scale magnetic field dynamo is not observed. In a mirror axionic soliton, a phase transition, which produces a magnetic helical flow, is possible. Using this transition, the soliton becomes mirror-asymmetric. When the mirror symmetry is broken, the axion soliton allows the magnetic energy, which is the result of the transformation of the axionic energy. In the main result, for an initial stage of the process, we calculate a scale for which the generation of large scale magnetic fields is the most intense. By making numerical simulations, we received that lower lateral harmonics of the magnetic field have greater amplitudes compared to higher ones. A simplest statistical ensemble, which is defined by the projection of all harmonics onto principal harmonics is constructed. We put forward an assumption that it was the indication to some instability in axionic MHD. Now, we can provide a possible explanation of this feature. When the mirror symmetry of the axion soliton is broken, the $\gamma$-term in the axionic mean field equation interacts with principal harmonics. As the result, the axion soliton acquires the magnetic energy and becomes helical. Read complete

Simone Fabbri, Alessandro Giuliani, Robin Reuvers

The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and the `topological' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of $e^2/h$: this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities. Read complete

Andrei G. Pronko

Motzkin chain is a model of nearest-neighbor interacting quantum $s=1$ spins with open boundary conditions. It is known that it has a unique ground state which can be viewed as a sum of Motzkin paths. We consider the case of periodic boundary conditions and provide several conjectures about structure of the ground state space and symmetries of the Hamiltonian. We conjecture that the ground state is degenerate and independent states distinguished by the eigenvalue of the third component of the total spin operator. Each of these states can be described as a sum of paths, similar to the Motzkin paths. Moreover, there exist two operators commuting with the Hamiltonian, which play the roles of lowering and raising operators when acting at these states. We conjecture also that these operators generate the Lie algebra of $C$-type of the rank equal to the number of sites. The symmetry algebra of the Hamiltonian is actually wider, and extended, besides the cyclic shift operator, by a central element contained in the third component of the total spin operator. Read complete

Mireille Bousquet-Mélou, Ariane Carrance, Baptiste Louf

We design a recursive algorithm to compute the partition function of the Ising model, summed over cubic maps with fixed size and genus. The algorithm runs in polynomial time, which is much faster than methods based on a Tutte-like, or topological, recursion. We construct this algorithm out of a partial differential equation that we derive from the first equation of the KP hierarchy satisfied by the generating function of bipartite maps. This series is indeed related to the Ising partition function by a change of variables. We also obtain inequalities on the coefficients of this partition function, which should be useful for a probabilistic study of cubic Ising maps whose genus grows linearly with their size. Read complete

Piotr Kopszak, Dmitry Grinko, Adam Burchardt, Maris Ozols, Michał Studziński, Marek Mozrzymas

We study entangled resource state recycling after one round of probabilistic port-based teleportation. We analytically characterize its degradation and, for the case of the resource state consisting of $N$ EPR pairs, we demonstrate the possibility of reusing it for a subsequent round of teleportation in the $N \to \infty$ limit. For the case of an optimized resource state, we compare the protocol's performance to multi-port-based teleportation, indicating that the resource state reuse is possible. Read complete