Finite-dimensional Lattices Research Articles

A note on the Banach lattice $c_0( \ell_2^n)$, its dual and its bidual

The main purpose of this paper is to study some geometric and topological properties of $c_0$-sum of the finite dimensional Banach lattice $\ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattice $c_0(\ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\infty}(\ell_2^n)$ is an almost limited set.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Transfer matrices for discrete Hermitian operators and absolutely continuous spectrum

We introduce a transfer matrix method for the spectral analysis of discrete Hermitian operators with locally finite hopping. Such operators can be associated with a locally finite graph structure and the method works in principle on any such graph. The key result is a spectral averaging formula well known for Jacobi or 1-channel operators giving the spectral measure at a root vector by a weak limit of products of transfer matrices. Here, we assume an increase in the rank for the connections between spherical shells which is a typical situation and true on finite dimensional lattices Zd. The product of transfer matrices are considered as a transformation of the relations of ‘boundary resolvent data’ along the shells. The trade off is that at each level or shell with more forward than backward connections (rank-increase) we have a set of transfer matrices at a fixed spectral parameter. Still, considering these products we can relate the minimal norm growth over the set of all products with the spectral measure at the root and obtain several criteria for absolutely continuous spectrum. Finally, we give some example of operators on stair-like graphs (increasing width) which has absolutely continuous spectrum with a sufficiently fast decaying random shell-matrix-potential.

Sharp tunneling estimates for a double-well model in infinite dimension

We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the L2 spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension.

Improved local spectral gap thresholds for lattices of finite size

Knabe's theorem lower bounds the spectral gap of a one dimensional frustration-free local hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap much scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free hamiltonians on a finite dimensional lattice, that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of coarse-grained hamiltonian (introduced in [Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.

Synchronization of discrete oscillators on ring lattices and small-world networks

A lattice of three-state stochastic phase-coupled oscillators exhibits a phase transition at a critical value of the coupling parameter a, leading to stable global oscillations. On a complete graph, upon further increase in a, the model exhibits an infinite-period (IP) phase transition, at which collective oscillations cease and discrete rotational (C3) symmetry is broken. The IP phase does not exist on finite-dimensional lattices. In the case of large negative values of the coupling no synchronization is expected, but nonetheless it was shown that travelling-wave steady states are stable, displaying local order (Escaff et al 2014 Phys. Rev. E 90 052111). Here, we verify the IP phase in systems with long-range coupling but of lower connectivity than a complete graph and show that even for large positive coupling, the system sometimes fails to reach global order. The ensuing travelling-wave state appears to be a metastable configuration whose birth and decay (into the previously described phases) are associated with the initial conditions and fluctuations.

Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions

For finite dimensional hermitean inner product spaces V, over \(*\)-fields F, and in the presence of orthogonal bases providing form elements in the prime subfield of F, we show that quantifier-free definable relations in the subspace lattice \(\mathsf{L}(V)\), endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F, also in terms of Grassmann–Plucker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type \(\Sigma _1\). In absence of involution, these results remain valid.

Lattice vector spaces and linear transformations

This paper introduces the concept of lattice vector space and establishes many important results. Also, this paper deals with linear transformations on lattice vector spaces and discusses their elementary properties. We prove that every finite dimensional lattice vector space is isomorphic to [Formula: see text] and show that the set of all columns (or the set of all rows) of an invertible matrix over [Formula: see text] is a basis for [Formula: see text].

Emergence of linguistic conventions in multi-agent reinforcement learning.

Recently, emergence of signaling conventions, among which language is a prime example, draws a considerable interdisciplinary interest ranging from game theory, to robotics to evolutionary linguistics. Such a wide spectrum of research is based on much different assumptions and methodologies, but complexity of the problem precludes formulation of a unifying and commonly accepted explanation. We examine formation of signaling conventions in a framework of a multi-agent reinforcement learning model. When the network of interactions between agents is a complete graph or a sufficiently dense random graph, a global consensus is typically reached with the emerging language being a nearly unique object-word mapping or containing some synonyms and homonyms. On finite-dimensional lattices, the model gets trapped in disordered configurations with a local consensus only. Such a trapping can be avoided by introducing a population renewal, which in the presence of superlinear reinforcement restores an ordinary surface-tension driven coarsening and considerably enhances formation of efficient signaling.

Definable relations in finite-dimensional subspace lattices with involution

For a large class of finite dimensional inner product spaces V, over division \(*\)-rings F, we consider definable relations on the subspace lattice \(\mathsf{L}(V)\) of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the \(*\)-ring \(\mathsf{R}(V)\) of endomorphisms of V, thereby identifying \(\mathsf{L}(V)\) with the lattice of right ideals of \(\mathsf{R}(V)\), with the induced involution. As an application, model completeness of F is shown to imply that of \(\mathsf{R}(V)\) and \(\mathsf{L}(V)\).

On the countable orthogonal lifting property

We proceed from Conrad’s counterexample to countable orthogonal lifting which disproved a positive result by Topping when working over the real field. It is shown that Conrad’s work applies more generally, beginning with an abelian group which is a lower semi-lattice that also contains an element greater than 0. Conrad’s example was of uncountable dimension. It is shown that Conrad-type vector space examples of countable dimension exist when working with a field whose elements are linearly ordered, in particular with the real field. An example shows the failure of the orthogonal lifting property in a finite dimensional vector lattice over the two-element field.

Transverse spin correlations of the random transverse-field Ising model

The critical behavior of the random transverse-field Ising model in finite dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$ and $3$ dimensions. At the critical point an algebraic decay of the form $\sim r^{-\eta_t}$ is found, with a decay exponent being approximately $\eta_t \approx 2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random AF XX-chain and have been tested by numerical calculations using free-fermionic techniques.

Comparison of Gabay–Toulouse and de Almeida–Thouless instabilities for the spin-glass XY model in a field on sparse random graphs

Vector spin glasses are known to show two different kinds of phase transitions in presence of an external field: the so-called de Almeida-Thouless and Gabay-Toulouse lines. While the former has been studied to some extent on several topologies (fully connected, random graphs, finite-dimensional lattices, chains with long-range interactions), the latter has been studied only in fully connected models, which however are known to show some unphysical behaviors (e.g. the divergence of these critical lines in the zero-temperature limit). Here we compute analytically both these critical lines for XY spin glasses on random regular graphs. We discuss the different nature of these phase transitions and the dependence of the critical behavior on the field distribution. We also study the crossover between the two different critical behaviors, by suitably tuning the field distribution.

Constraint percolation on hyperbolic lattices.

Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices, and they are interesting in their own right, with ordinary percolation exhibiting not one but two phase transitions. We study four constraint percolation models-k-core percolation (for k=1,2,3) and force-balance percolation-on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggest that all of the k-core models, even for k=3, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the k-core percolation models.

Self-sustained clusters as drivers of computational hardness in p -spin models

While macroscopic properties of spin glasses have been thoroughly investigated, their manifestation in the corresponding microscopic configurations is much less understood. Cases where both descriptions have been provided, such as constraint satisfaction problems, are limited to their ground state properties. To identify the emerging microscopic structures with macroscopic phases at different temperatures, we study the $p$-spin model with $p\!=\!3$. We investigate the properties of self-sustained clusters, defined as variable sets where in-cluster induced fields dominate over the field induced by out-cluster spins, giving rise to stable configurations with respect to fluctuations. We compute the entropy of self-sustained clusters as a function of temperature and their sizes. In-cluster fields properties and the difference between in-cluster and out-cluster fields support the observation of slow-evolving spins in spin models. The findings are corroborated by observations in finite dimensional lattices at low temperatures.

Efficiently Controllable Graphs.

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that, when a controllable quantum network is described by such a graph and the gaps in eigenfrequencies and in transition frequencies are bounded exponentially in the number of vertices, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficiently controllable and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems or qubits.

Feasibility Analysis on the Attenuation of Strong Ground Motions Using Finite Periodic Lattices of Mass-in-Mass Barriers

AbstractThis study assesses the implementation of locally resonant metamaterials in seismic isolation applications by investigating their potential feasibility in the [0.5, 5.0] Hz frequency band. To this end, via adoption of both Bloch’s theory and classical vibration analysis, the one-dimensional mass-in-mass lattice is analyzed in three overlapping subbands and corresponding relations for the structural parameters are derived. Accordingly, a basic unit cell is proposed that incorporates a core of dense material that is suspended in a thin shell composed of a lightweight material using distributed elastomeric tendons. Initial numerical simulations on finite-dimensional lattices reveal significant potential in the proposed solution and encourage further investigation, particularly to the two-dimensional case.

Asymptotic Behavior of Factorization and Projection Constants

This paper is concerned with estimates of important factorization constants that appear in Banach space theory. We prove upper bounds of the Hilbertian norm of projections on finite-dimensional spaces of interpolation spaces generated by certain abstract interpolation functors and show applications to Calderon–Lozanovskii spaces. We also prove estimates of the p-factorization norm and projection constants for finite-dimensional Banach lattices. We show as a consequence of our results that in a large class of n-dimensional Banach sequence lattices \(E_n\) the projection constants \(\lambda (E_n)\) satisfy \(\lim _{n\rightarrow \infty }\lambda (E_n)/\sqrt{n} = c\), where \(c=\sqrt{2/\pi }\) in the real case and \(c= \sqrt{\pi }/2\) in the complex case. Applications are given to vector-valued sequence spaces.

Loop-corrected belief propagation for lattice spin models

Belief propagation (BP) is a message-passing method for solving probabilistic graphical models. It is very successful in treating disordered models (such as spin glasses) on random graphs. On the other hand, finite-dimensional lattice models have an abundant number of short loops, and the BP method is still far from being satisfactory in treating the complicated loop-induced correlations in these systems. Here we propose a loop-corrected BP method to take into account the effect of short loops in lattice spin models. We demonstrate, through an application to the square-lattice Ising model, that loop-corrected BP improves over the naive BP method significantly. We also implement loop-corrected BP at the coarse-grained region graph level to further boost its performance.

Vertex-transitive median graphs of non-exponential growth

We characterize vertex-transitive median graphs of non-exponential growth as the Cartesian products of finite hypercubes with finite dimensional lattice graphs. Additionally, we prove that every median graph without convex subgraphs isomorphic to K1,3 or the 4-pan graph is isomorphic to the weak Cartesian product of finite paths, rays and two way infinite paths.