Infinite Trees Research Articles

Variable Lebesgue Space over Weighted Homogeneous Tree

An infinite homogeneous tree is a special type of graph that has a completely symmetrical structure in all directions. For an infinite homogeneous tree T=(V,E) with the natural distance d defined on graphs and a weighted measure μ of exponential growth, the authors introduce the variable Lebesgue space Lp(·)(μ) over (V,d,μ) and investigate it under the global Hölder continuity condition for p(·). As an application, the strong and weak boundedness of the maximal operator relevant to admissible trapezoids on Lp(·)(μ) is obtained, and an unbounded example is presented.

$$A_p$$ weights on nonhomogeneous trees equipped with measures of exponential growth

AbstractThis paper aims to study $$A_p$$ A p weights in the context of a class of metric measure spaces with exponential volume growth, namely infinite trees with root at infinity equipped with the geodesic distance and flow measures. Our main result is a Muckenhoupt Theorem, which is a characterization of the weights for which a suitable Hardy–Littlewood maximal operator is bounded on the corresponding weighted $$L^p$$ L p spaces. We emphasise that this result does not require any geometric assumption on the tree or any condition on the flow measure. We also prove a reverse Hölder inequality in the case when the flow measure is locally doubling. We finally show that the logarithm of an $$A_p$$ A p weight is in BMO and discuss the connection between $$A_p$$ A p weights and quasisymmetric mappings.

Equable parallelograms on the Eisenstein lattice

Abstract This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger’s Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose vertices have degree 4, bar the root which has degree 3. This study naturally complements the authors’ previous study of equable parallelograms whose vertices lie on the integer lattice.

Checking equivalence of corecursive streams: An inductive procedure

In recent work, non-periodic streams have been defined corecursively, by representing them with finitary equational systems built on top of various operators, besides the standard constructor. When only the stream constructor is allowed in equations, only periodic streams can be represented, and the structures of periodic streams and infinite regular trees are isomorphic. Therefore, one can use the theory of regular trees to get a sound and complete procedure to decide whether two equational systems are equivalent, that is, define the same streams. However, such an isomorphism no longer exists if one allows other operators in equations; in particular, there exist systems of equations which have the same unique solution as streams, but not as regular trees. Hence, equality of regular trees becomes stronger then equality of streams, with a negative impact on termination of functions whose definition is based on the equivalence of the representation of streams as finitary equational systems. To overcome this problem, we provide a weaker definition of equivalence, and prove its soundness and relative completeness. This definition is coinductive, hence non-algorithmic. However, we show that it can be turned into an equivalent inductive procedure.

The spectrum of asymptotic Cayley trees

We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.

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The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\beta >0$\\end{document} per edge. It arises as the q→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\ o 0$\\end{document} limit of the q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document}-state random cluster model with p=βq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p=\\beta q$\\end{document}. We prove that in dimensions d⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d\\geqslant 3$\\end{document} the arboreal gas undergoes a percolation phase transition. This contrasts with the case of d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d=2$\\end{document} where no percolation transition occurs.The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane H0|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{H}^{0|2}$\\end{document}. This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the H0|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{H}^{0|2}$\\end{document} model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

Anomalous criticality coexists with giant cluster in the uniform forest model

We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant-tree size displays two-length scaling fluctuations, instead of following the central-limit theorem.

Top-down complementation of automata on finite trees

We present a new complementation construction for nondeterministic automata on finite trees. The traditional complementation involves determinization of the corresponding bottom-up automaton (recall that top-down deterministic automata are less powerful than nondeterministic automata, whereas bottom-up deterministic automata are equally powerful).The construction works directly in a top-down fashion, therefore without determinization. The main advantages of this construction are: (i) in the special case of finite words it boils down to the standard subset construction (which is not the case of the traditional bottom-up complementation construction), and (ii) it illustrates the core argument of the complementation lemma for infinite trees, central in the proof of Rabin's tree theorem, in a simpler setting where issues related to acceptance conditions over infinite words and determinacy of infinite games are not present.

Sublattice-selective percolation on bipartite planar lattices.

In conventional site percolation, all lattice sites are occupied with the same probability. For a bipartite lattice, sublattice-selective percolation instead involves two independent occupation probabilities, depending on the sublattice to which a given site belongs. Here, we determine the corresponding phase diagram for the two-dimensional square and Lieb lattices from quantifying the parameter regime where a percolating cluster persists for sublattice-selective percolation. For this purpose, we present an adapted Newman-Ziff algorithm. We also consider the critical exponents at the percolation transition, confirming previous Monte Carlo and renormalization-group findings that suggest sublattice-selective percolation belongs to the same universality class as conventional site percolation. To further strengthen this conclusion, we finally treat sublattice-selective percolation on the Bethe lattice (infinite Cayley tree) by an exact solution.

Factorisation in the semiring of finite dynamical systems

Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS A is cancellative if one can divide by it unambiguously, i.e. AX=AY implies X=Y. We prove that an FDS A is cancellative if and only if it has a fixed point. Secondly, we prove that if an FDS A has a k-th root (i.e. B such that Bk=A), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs.

Tipsy cop and tipsy robber: Collisions of biased random walks on graphs

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the tipsy cop and drunken robber is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop moves directed towards the robber some fixed proportion of the time and randomly the remainder. In this article, we adopt a slightly different interpretation of tipsiness of the cop and robber where we assume that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly random) move by the robber. We study this tipsy cop and tipsy robber game on the infinite grid graph and on certain families of infinite trees including δ-regular trees and δ-regular trees rooted to a Δ-regular tree, where Δ≥δ. Our main results analyze strategies for the cop and robber on these graphs. We conclude with some directions for further study.

Extensions and reductions of squarefree words

A word is squarefree if it does not contain a nonempty word of the form XX as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long squarefree words over a 3-letter alphabet. We study squarefree words with additional properties involving single-letter deletions and extensions of words.A squarefree word is steady if it remains squarefree after deletion of any single letter. We prove that there exist infinitely many steady words over a 4-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size 7 assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to 4, which is best possible.In the opposite direction, we consider squarefree words that remain squarefree after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly-infinite bifurcate words over an alphabet of size 12.

Two-weight dyadic Hardy inequalities

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research.

Deterministic rendezvous in infinite trees

The rendezvous task calls for two mobile agents, starting from different nodes of a network modeled as a graph to meet at the same node. Agents have different labels which are integers from a set {1,…,L}. They wake up at possibly different times and move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous algorithms. The time of such an algorithm is the number of rounds since the wakeup of the earlier agent till the meeting. In most of the literature concerning rendezvous in graphs, the graph is finite and the time of rendezvous depends on its size. This approach is impractical for very large graphs and impossible for infinite graphs. For such graphs it is natural to design rendezvous algorithms whose time depends on the initial distance D between the agents. In this paper we adopt this approach and consider rendezvous in infinite trees. All our algorithms work in finite trees as well. Our main goal is to study the impact of orientation of a tree on the time of rendezvous.We first design a rendezvous algorithm working for unoriented regular trees, whose time is in O(z(D)log⁡L), where z(D) is the size of the ball of radius D, i.e., the number of nodes at distance at most D from a given node. The algorithm works for arbitrary delay between waking times of agents and does not require any initial information about parameters L or D. Its disadvantage is its complexity: z(D) is exponential in D for any degree d>2 of the tree. We prove that this high complexity is inevitable: Ω(z(D)) turns out to be a lower bound on rendezvous time in unoriented regular trees, even for simultaneous start and even when agents know L and D. Then we turn attention to oriented trees. While for arbitrary delay between waking times of agents the lower bound Ω(z(D)) still holds, for simultaneous start the time of rendezvous can be dramatically shortened. We show that if agents know either a polynomial upper bound on L or a linear upper bound on D, then rendezvous can be accomplished in oriented trees in time O(Dlog⁡L), which is optimal. If no such extra knowledge is available, a significant speedup is still possible: in this case we design an algorithm working in time O(D2+log2⁡L).

Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects.

It is known that maximal entropy random walks and partition functions that count long paths on graphs tend to become localized near nodes with a high degree. Here, we revisit the simplest toy model of such a localization: a regular tree of degree p with one special node ("root") that has a degree different from all the others. We present an in-depth study of the path-counting problem precisely at the localization transition. We study paths that start from the root in both infinite trees and finite, locally tree-like regular random graphs (RRGs). For the infinite tree, we prove that the probability distribution function of the endpoints of the path is a step function. The position of the step moves away from the root at a constant velocity v=(p-2)/p. We find the width and asymptotic shape of the distribution in the vicinity of the shock. For a finite RRG, we show that a critical slowdown takes place, and the trajectory length needed to reach the equilibrium distribution is on the order of N instead of logp-1N away from the transition. We calculate the exact values of the equilibrium distribution and relaxation length, as well as the shapes of slowly relaxing modes.

Rao-Fisher information geometry and dynamics of the event-universe views distributions

We developed the novel mathematical model for event-universe by representing events as branches of dendrograms (finite trees) expressing the hierarchic relation between events. At the ontic level we operate with infinite trees. Algebraically such mathematical structures are represented as p-adic numbers. We call this kind of event mechanics Dendrogramic Holographic theory (DHT). It can be considered as a fundamental theory generating both GR and QM. In this paper we endower DHT with Rao-Cramer's information geometry. Following Smolin's derivation of QM from the event-universe, we introduce views from one event to others and by using their probability distributions we invent stochastic geometry. The important mathematical result is that all such views' distributions can be parametrized by four real parameters that are a part of the shape complexity measure introduced by Barbour in his particle shape dynamics theory – adapted to DHT. Hence, within DHT all possible event-universes can be embedded in four-dimensional real space. Asin GR, we introduce proper time. This “proper time” depends only on the change between one distribution of an observer to the other. The linkage of time to change is highlighted in the ideology of Rovelli and Barbour's shape dynamics.

Infinite constant gap length trees in products of thick Cantor sets

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to be countably infinite, which further distinguishes this work from previous results on patterns in fractal sets. This builds on the authors’ previous work on graphs and distance sets over products of Cantor sets of sufficient Newhouse thickness.

Unfoldings and Coverings of Weighted Graphs

Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings of finite graphs. We prove similar results for unfoldings of finite directed graphs. Moreover, we generalize coverings and similarly, unfoldings to graphs and digraphs equipped with finite or infinite weights attached to edges of the covered or unfolded graphs. This generalization yields a canonical “factorization” of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing ω as an infinite weight provides us with finite descriptions of regular trees having nodes of countably infinite degree. Regular trees (trees having finitely many subtrees up to isomorphism) play an important role in the extension of Formal Language Theory to infinite structures described in finitary ways. Our weighted graphs offer effective descriptions of the above mentioned regular trees and yield decidability results. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.

Representation of the universe as dendrogramic hologram empowered with relational interpretation

This is a brief review on the basics of recently established Dendrogramic Holographic theory (DH-theory). This is the special model of the event-universe based on the clustering transformation of experimental data into dendrogram, a finite tree which branches encoding the events. These event-branches are coupled via the hierarchic interrelation determined by the dendrogram. Such relational universe differs from the universe with space-time mathematically described by the real numbers. Dendrogram is endowed with the common root ultrametric. Finite dendrograms correspond to the epistemic level of description; in the limit we obtain an infinite tree providing the ontic description. In the simplest model, the tree is homogeneous, p-adic tree. It can be endowed with the algebraic structure of the ring of p-adic integers. Hence, DH-theory is a part (but very special) of p-adic theoretical physics. In this paper we discuss the foundations of DH-theory and its applications to quantum-classical interrelation including the novel interpretation of the violations of the CHSH-inequality, to general relativity, and to emergence of quantum mechanics from the event-picture of the universe. Since both quantum theory and general relativity can be emergent from DH-theory, creation of the latter can be viewed as a step towards unification of these two fundamental physical theories.

Uniform Restricted Chase Termination

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A central problem concerning the chase procedure is uniform (a.k.a. all-instances) chase termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is, in general, undecidable, it is natural to ask whether known well-behaved classes of TGDs ensure decidability. We focus on the main paradigms that led to robust TGD-based formalisms, namely guardedness and stickiness, that have been introduced in the context of knowledge-enriched databases. Although uniform chase termination is well understood for the oblivious version of the chase (2EXPTIME-complete for guarded, and PSPACE-complete for sticky TGDs), the more subtle case of the restricted (a.k.a. the standard) chase is rather unexplored. We show that uniform restricted chase termination under guarded single-head TGDs and sticky single-head TGDs is decidable in elementary time. In the case of guardedness, we provide a reduction to the satisfiability problem of monadic second-order logic over infinite trees of bounded degree, while for stickiness we provide a reduction to the emptiness problem of deterministic Büchi automata. Those reductions build on a series of technical results of independent interest related to the notion of fairness of the restricted chase, and the existence of critical databases that characterize nontermination of the restricted chase via databases of a certain form.