Equilibrium moderate deviations for occupation times of SSEP on regular trees

Abstract What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non‐backtracking random walk. This generalizes the well‐known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree.

or the effective involvement of cultural heritage sites in the tourism sector, the preservation of historical memory, and the implementation of comprehensive restoration work, studies aimed at identifying the range of plants, the spatial-planning structure of green spaces and assessing their current state are of particular relevance. The purpose of the study is to assess the current state of green spaces on the territory of the former Kologrivsky Lower Agricultural Technical School named after F.V. Chizhov (Kostroma Region). The study used a set of methods: historical and archival analysis, field surveys and a graphoanalytical method. During field studies in 2023-2025, an inventory of green spaces was carried out, their sanitary condition was assessed, and the species composition of undercanopy plants was studied. A complex spatial-planning structure of the territory was revealed, including a linden alley and a landscape park with fragments of regular planning and trees aged 180-190 years. The tree plantations are in a weakened state, and the share of drying and dead trees reaches 26% in the alley and 35% in the park. Successful natural regeneration of linden and elm is noted under the canopy of the park. In general, a decrease in biodiversity and a threat of losing the preserved historical assortment of trees and shrubs are predicted in the coming decades. The widespread distribution of Sosnowsky's hogweed on the territory of the complex and the use of chemical methods of controlling it create a threat of physical loss of the park landscape. The results obtained in the course of the study indicate the need to develop and implement a set of conservation and restoration measures, including forestry ones for the care of plantations and improving their sanitary condition. The study results form a scientific foundation for developing preservation and restoration measures, applicable not only to this particular object but also to other provincial historical parks in central Russia facing similar conditions.

Abstract In this article we study the theories of the infinite-branching tree and the r -regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of graphs, and provide a characterization of the Morley rank of definable sets in terms of the degrees of polynomials measuring their non-standard cardinalities. This answers negatively some questions from [2], where it is asked whether every stable generalised measurable structure is one-based.

The article focuses on a class of second countable groups assembled from profinite and discrete groups by elementary operations. We focus on a rank associated with these groups that measures their complexity, the decomposition rank. A collection of groups acting on \aleph_{0} -regular trees is defined and used for the first construction of a group with decomposition rank \omega^{\omega}+1 .

Pruning is a regular and essential urban tree maintenance practice aimed at sustaining overall health, ecosystem services, and public safety. However, knowledge of post-pruning recovery dynamics remains limited, which in turn hinders accurate assessments of growth and ecological functions. To address this, we examined recovery dynamics of Acacia auriculiformis, a common urban species. Tree height and crown radius were recorded monthly for 12 months after pruning. Trees were classified into two size groups based on diameter at breast height (DBH, trunk diameter measured at 1.3 m above ground): medium (DBH < 45 cm) and large (DBH ≥ 45 cm). A generalized linear mixed model (GLMM), appropriate for repeated measures and non-normal data, was fitted using a Tweedie distribution and a log-link function to model the recovery pattern. Results showed continuous growth over time, with medium-sized trees presenting significantly higher crown radius growth than large trees (p = 0.006), while height growth did not differ (p = 0.788). The best model for height included time (AIC = −846.4), whereas crown recovery was best modelled by time and size class (AIC = −1586.6). These findings demonstrate that, in this study, medium-sized A. auriculiformis generally recover faster, especially in crown expansion. This exploratory study suggests that tree size may influence post-pruning recovery and can provide a reference for subsequent differentiated management studies. The morphological modeling further provides preliminary quantitative evidence for annual recovery dynamics in urban A. auriculiformis.

Non-Gibbsian Multivariate Ewens Probability Distributions on Regular Trees

Abstract We consider $$\mathbb {Z}_q$$ Z q -valued clock models on a regular tree, for general classes of ferromagnetic nearest neighbor interactions which have a discrete rotational symmetry. It has been proved recently that, at strong enough coupling, families of homogeneous Markov chain Gibbs states $$\mu _A$$ μ A coexist whose single-site marginals concentrate on $$A\subset \mathbb {Z}_q$$ A ⊂ Z q , and which are not convex combinations of each other [1]. In this note, we aim at a description of the extremal decomposition of $$\mu _A$$ μ A for $$|A|\ge 2$$ | A | ≥ 2 into all extremal Gibbs measures, which may be spatially inhomogeneous. First, we show that in regimes of very strong coupling, $$\mu _A$$ μ A is not extremal. Moreover, $$\mu _A$$ μ A possesses a single-site reconstruction property which holds for spin values sent from the origin to infinity, when these initial values are chosen from A . As our main result, we show that $$\mu _A$$ μ A decomposes into uncountably many extremal inhomogeneous states. The proof is based on multi-site reconstruction which allows to derive concentration properties of branch overlaps. Our method is based on a new good site/bad site decomposition adapted to the A -localization property, together with a coarse graining argument in local state space.

Northern forests are currently taking up large quantities of carbon due to tree growth. Yet, it is not known how the stocks of soil nutrients have responded to this increase in biomass. Therefore, we analyzed thousands of forest soils in Sweden over the last four decades. We found strong increases in the concentrations and stocks of plant-available magnesium, calcium, and manganese in the organic layer of the soils. Specifically, the concentrations of plant-available magnesium, calcium, and manganese in the organic layer increased by 38%, 21%, and 100%, respectively, over the four decades. These increases were related to soil texture and the magnesium concentration of the soil parent material as well as to the dominant tree species. The increase in nutrients in the organic layer might be caused by an uplift of nutrients from the mineral subsoil to the organic layer due to plant nutrient uptake in the subsoil and litter fall and they might also be driven by decreased leaching of nutrients from the organic layer due to decreased acid deposition. Concurrently, the nitrogen content of the organic layer decreased over the four decades. In conclusion, our results show that stocks and concentrations of plant-available cations in the organic layer of Swedish forest soils increased despite increases in tree biomass and regular tree harvests. Our study indicates that there is a low risk for base cation deficiency and that nitrogen will remain the limiting nutrient for tree growth in northern forests.

We study scalar field theory on biregular trees, as a new model for discrete holography. Biregular trees are discrete symmetric spaces associated with the bulk isometry group SU ( 3 ) over the unramified quadratic extension of a nonarchimedean field. The bulk-to-bulk and bulk-to-boundary propagators exhibit distinct features absent on the regular tree or continuum AdS spaces, arising from the semihomogeneous nature of the bulk space. We compute the two- and three-point correlators of the putative boundary dual. The three-point correlator exhibits a nontrivial “tensor structure” via dependence on the homogeneity degree of a unique bulk point specified in terms of boundary insertion points. The computed operator product expansion coefficients show dependence on zeta functions associated with the unramified quadratic extension of a nonarchimedean field. This work initiates the formulation of holography on a family of discrete holographic spaces that exhibit features of both flat space and negatively curved space.

We investigate the entanglement structure of a generic MM-particle Bethe wavefunction (not necessarily an eigenstate of an integrable model) on a 1d lattice by dividing the lattice into LL parts and decomposing the wavefunction into a sum of products of LL local wavefunctions. Using the fact that a Bethe wavefunction accepts a fractal multipartite decomposition – it can always be written as a linear combination of L^MLM products of LL local wavefunctions, where each local wavefunction is in turn also a Bethe wavefunction – we then build exact, analytical tensor network representations with finite bond dimension \chi=2^Mχ=2M, for a generic planar tree tensor network (TTN), which includes a matrix product states (MPS) and a regular binary TTN as prominent particular cases. For a regular binary tree, the network has depth \log_{2}(N/M)log2(N/M) and can be transformed into an adaptive quantum circuit of the same depth, composed of unitary gates acting on 2^M2M-dimensional qudits and mid-circuit measurements, that deterministically prepares the Bethe wavefunction. Finally, we put forward a much larger class of generalized Bethe wavefunctions, for which the above decompositions, tensor network and quantum circuit representations are also possible.

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the ( d + 1 ) (d+1) -regular tree T ~ d + 1 \widetilde T_{d+1} , where d d is the residual degree. Then we introduce the class of liftable self-similar groups acting on a d d -regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on T ~ d + 1 \widetilde T_{d+1} fixing one of the ends. The closures of these extensions in A u t ( T ~ d + 1 ) Aut(\widetilde T_{d+1}) are totally disconnected locally compact groups that belong to the class of scale groups as defined by Willis [Scale groups, 2022]. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely on the boundaries of rooted trees or groups admitting finite L L -presentations. In particular, we show that the finitely presented group constructed by the first author [Mat. Sb. 189 (1998), pp. 79–100] and the finitely presented HNN extension of the Basilica group constructed by Grigorchuk and Żuk [Spectral properties of a torsion-free weakly branch group defined by a three state automaton, Amer. Math. Soc., Providence, RI, 2002, pp. 57–82] embed into the group D ( Q 2 ) \mathcal D(\mathbb {Q}_2) of dilations of the field Q 2 \mathbb {Q}_2 of 2 2 -adic numbers. These actions, translated to T ~ 3 \widetilde T_3 , are 2-transitive on the punctured boundary of T ~ 3 \widetilde T_3 . Also we explore scale-invariant groups studied by Nekrashevych and Pete [Groups Geom. Dyn. 5 (2011), pp. 139–167] with the purpose of getting new examples of scale groups.

Hyperuniformity in Regular Trees

Abstract It is a well-known result due to Bollobás that the maximal Cheeger constant of large $d$ d -regular graphs cannot be close to the Cheeger constant of the $d$ d -regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by $2/\pi \approx 0.63$ 2 / π ≈ 0.63 ... which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson–Voronoi tessellation of the surface with a vanishing intensity.

Abstract Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for all constants $b,\alpha>0$ and $1<\lambda < d-1$, there exist constants $c,C>0$ such that the following holds. Let $G$ be a $d$-regular graph on $n$ vertices, satisfying that for every $U\subseteq V(G)$ with $|U|\le \frac{n}{2}$, $e(U,U^{c})\ge b|U|$ and for every $U\subseteq V(G)$ with $|U|\le \log ^{C}n$, $e(U)\le (1+c)|U|$. Let $p=\frac{\lambda }{d-1}$. Then, with probability tending to one as $n$ tends to infinity, the largest component $L_{1}$ in the random subgraph $G_{p}$ of $G$ satisfies $\left |1-\frac{|L_{1}|}{yn}\right |\le \alpha $, and all the other components in $G_{p}$ are of order $O\left (\frac{\lambda \log n}{(\lambda -1)^{2}}\right )$. This generalises (and improves upon) results for random $d$-regular graphs.

Abstract In this note we study the Landis conjecture for positive Schrödinger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when $ \mathcal{H} $-harmonic functions for a positive Schrödinger operator $ \mathcal{H} $ with potentials bounded from above by $ 1 $ are trivial. The positivity assumption on the operator allows us to impose slow decay across the entire graph, while requiring fast decay in only one direction, rather than throughout the whole graph. We then specifically look at the special cases of $ \mathbb{Z}^{d} $ and regular trees for which we get an explicit decay criterion. Moreover, we consider the fractional analogue of the Landis conjecture on $ \mathbb{Z}^{d} $. Our approach relies on the discrete version of Liouville comparison principle which is also proved in this article.

Quantum probability approaches for regular tree

Abstract Multicollinearity and heterogeneity are prevalent challenges in the analysis of process industry datasets, necessitating algorithms capable of addressing both simultaneously. The partial least squares (PLS)‐Tree algorithm, which integrates PLS regression with decision tree methodologies, stands out by concurrently addressing data heterogeneity and improving predictive performance. However, the PLS‐Tree algorithm remains underexplored compared to other machine learning approaches. This study delves into the intricacies of the PLS‐Tree algorithm, utilizing synthetic data that mirrors the complexity of real‐world process industry scenarios characterized by high collinearity and clustering. This paper further enhances the original PLS‐Tree framework by introducing multiple latent score vectors, with the objective of refining the clustering process and boosting predictive accuracy beyond that of standard PLS and regression tree algorithms. Additionally, a comparative analysis is presented, evaluating the performance of the enhanced PLS‐Tree against regular PLS and regression tree, highlighting its potential for sophisticated data analysis in the process industries.

On regular trees defined from unfoldings and coverings

BackgroundInjuries due to falling coconuts are a common yet underreported form of trauma in the tropical regions. Although these might appear insignificant at first glance, the physical forces involved are potentially fatal. Despite their global prevalence, research on this subject remains scarce, making it a neglected public health concern. This study seeks to bridge these gaps by analysing the affected demographics, contributing factors and injury patterns. By enhancing the understanding of coconut fall-related injuries, this research seeks to create awareness about dangers of falling coconuts and inform the development of effective public health strategies to mitigate their impact.MethodsA retrospective study was conducted over a period of 3 years from January 2021 to December 2023 at a tertiary care centre on the southern coast of India. Patients who presented to emergency with coconut-fall related injuries were identified through a comprehensive review of nursing ledgers. Additional data including imaging, consultations & treatment details were retrieved from patient files and electronic medical records. Descriptive statistics of the recorded data like demographic variables, time of injury, injury patterns, injury severity score (ISS) and ED disposition were analysed by using Microsoft Excel 365.ResultsThe study population included 17 males and 12 females. Most patients were within the age group of 40–60, which comprised 48% of the total participants. The months of September and October reported the highest frequency of cases. Out of the 29 patients, 14 were farm workers who sustained coconut fall-related injuries. Injury patterns varied from mild soft tissue injuries to severe TBIs which include SDH and SAH. There were 3 patients who required surgery, and 7 patients were admitted. There were no fatalities reported, and average hospital stay was 4.5 days.ConclusionCoconut fall-related injuries in tropical regions is a significant but less recognised public health issue. Our study shows the necessity of seasonal preventive strategies, public awareness and safety measures for high-risk population like outdoor workers and older adults. Community focussed interventions, such as regular coconut tree pruning, installation of coconut safety nets and educational campaigns will help to reduce the incidence and severity of these injuries.