Uniform Spanning Research Articles

Resolvent of large random graphs

AbstractWe analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieltjes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdös‐Rényi graphs and graphs with a given degree sequence. We give examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Resonance characteristics of two-span continuous beam under moving high speed trains

The resonance characteristics of a two-span continuous beam traversed by moving high speed trains at a constant velocity is investigated, in which the continuous beam has uniform span length. Each span of the continuous beam is modeled as a Bernoulli-Euler beam and the moving trains are represented as a series of two degrees-of-freedom mass-springdamper systems at the axle locations. A method of modal analysis is proposed in this paper to investigate the vibration of two-span continuous beam. The effects of different influencing parameters, such as the velocities of moving trains, the damping ratios and the span lengths of the beam, on the dynamic response of the continuous beam are examined. The two-span continuous beam has two critical velocities causing two resonance responses, which is different from simple supported beam. The resonance condition of the two-span continuous beam is put forward which depends on the first and second natural frequency of the beam and the moving velocity.

The dimension of the SLE curves

Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8). Introduction. It has been conjectured by theoretical physicists that various lattice models in statistical physics (such as percolation, Potts model, Ising model, uniform spanning trees), taken at their critical point, have a continuous confor-mally invariant scaling limit when the mesh of the lattice tends to 0. Recently, Oded Schramm [15] introduced a family of random processes which he called Stochastic Loewner Evolutions (or SLE), that are the only possible conformally invariant scaling limits of random cluster interfaces (which are very closely related to all above-mentioned models). An SLE process is defined using the usual Loewner equation, where the driving function is a time-changed Brownian motion. More specifically, in the present paper we will be mainly concerned with SLE in the upper-half plane (sometimes called chordal SLE), defined by the following PDE:

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus

Let x and y be points chosen uniformly at random from \({\mathbb {Z}_n^4}\), the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n2(log n)1/6, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on \({\mathbb {Z}_n^4}\) is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for d ≥ 5, in combination with results of Lawler concerning intersections of four-dimensional random walks.

Ends in Uniform Spanning Forests

It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.

Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process

Let (G n ) =1 ∞ be a sequence of finite graphs, and let Y t be the length of a loop-erased random walk on G n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G n is the d-dimensional torus of size-length n for d≥4, the process (Y t ) =0 ∞ , suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous.

Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3

We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit mu of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure mu, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning tree measure on Z^d.

Euler Integrals for Commuting SLEs

Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a single positive parameter κ. As shown in, Ref. 8 the coexistence of n interfaces in such a domain implies algebraic (“commutation”) conditions. In the most interesting situations, the admissible laws on systems of n interfaces are parameterized by κ and the solution of a particular (finite rank) holonomic system.The study of solutions of differential systems, in particular their global behaviour, often involves the use of integral representations. In the present article, we provide Euler integral representations for solutions of holonomic systems arising from SLE commutation. Applications to critical percolation (general crossing formulae), Loop-Erased Random Walks (direct derivation of Fomin’s formulae in the scaling limit), and Uniform Spanning Trees are discussed. The connection with conformal restriction and Poissonized non-intersection for chordal SLEs is also studied.

Infinite Canonical Super-Brownian Motion and Scaling Limits

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on \(\mathbb{Z}^d\) when these objects are critical, mean-field and infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation above 6 dimensions, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

Abstract: We study the stationary distribution of the standard Abelian sandpile model in the box $Lam_n = [-n, n]^d cap d$ for $d ge 2$. We show that as $n o infty$, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in $d$. This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on $d$. In the case $d > 4$, we also make use of Wilson's method.

Minimum spanning trees and random resistor networks inddimensions

We consider minimum-cost spanning trees, both in lattice and Euclidean models, in d dimensions. For the cost of the optimum tree in a box of size L , we show that there is a correction of order L(theta) , where theta< or =0 is a universal d -dependent exponent. There is a similar form for the change in optimum cost under a change in boundary condition. At nonzero temperature T , there is a crossover length xi approximately T(-nu) , such that on length scales larger than xi, the behavior becomes that of uniform spanning trees. There is a scaling relation theta=-1/nu, and we provide several arguments that show that nu and -1/theta both equal nu(perc) , the correlation length exponent for ordinary percolation in the same dimension d , in all dimensions d> or =1 . The arguments all rely on the close relation of Kruskal's greedy algorithm for the minimum spanning tree, percolation, and (for some arguments) random resistor networks. The scaling of the entropy and free energy at small nonzero T , and hence of the number of near-optimal solutions, is also discussed. We suggest that the Steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions, as is the traveling salesman problem in two dimensions. Hence all will have the same value of theta=-3/4 in two dimensions.

Minimal spanning forests

Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In Z d this was proved by Alexander [Ann. Probab. 23 (1995) 87-104], but a different method is needed for the nonamenable case. We also prove that the WMSF components are thin in a different sense, namely, on any graph, each component tree in the WMSF has p c = 1 a.s., where p c denotes the critical probability for having an infinite cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be thick: on any connected graph, the union of the FMSF and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees). We also show that the number of infinite clusters for Bernoulli(p u ) percolation is at most the number of components of the FMSF, where p u denotes the critical probability for having a unique infinite cluster. Finally, an example is given to show that the minimal spanning tree measure does not have negative associations.

Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12,...

The uniform spanning forest (USF) in Z^d is the weak limit of random, uniformly chosen, spanning trees in [-n,n]^d. Pemantle proved that the USF consists a.s. of a single tree if and only if d = 9. More generally, let N(x,y) be the minimum number of edges outside the USF in a path joining x and y in Z^d. Then a.s. max{N(x,y) : x,y in Z^d} is the integer part of (d-1)/4. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.

Liquid-Handling Robotic Workstations for Functional Genomics

More and more functional genomics laboratories are willing to invest in robotic workstations due to the higher throughput liquid-handling intensive nature of the work. In this report, the features of robotic workstations important for functional genomics are discussed. Workstations for functional genomics are useful for replication of clone sets, PCR and sequencing set-up and clean-up, hit picking, gel loading, and nucleic acid purification procedures. Workstations not only increase throughput, but also ensure that assay steps are performed consistently, human error- and learning curve-free from run to run. Workstations fit on a laboratory bench and may have several robotic arms in different configurations. Available configurations include grippers and single-, oligo-, or multi-probe heads, with uniform or independent spanning, and liquid-level detection or tracking options. Most hardware and software packages can be designed for integration with other automated devices, building towards a fully automated system.

Negative association in uniform forests and connected graphs

AbstractWe consider three probability measures on subsets of edges of a given finite graph G, namely, those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is verified numerically for all graphs G having eight or fewer vertices, or having nine vertices and no more than 18 edges, using a certain computer algorithm which is summarized in this paper. Negative association is known already to be valid for a uniform spanning tree. The three cases of uniform forest, uniform spanning tree, and uniform connected subgraph are special cases of a more general conjecture arising from the random‐cluster model of statistical mechanics. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

We study the stationary distribution of the standard Abelian sandpile model in the box Λn = [-n, n] d ∩ ℤ d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ d . In the case d > 4, we also make use of Wilson’s method.

Couplings of uniform spanning forests

We prove the existence of an automorphism-invariant coupling for the wired and the free uniform spanning forests on connected graphs with residually amenable automorphism groups.

Conformal invariance of planar loop-erased random walks and uniform spanning trees

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $Dsubsetneqq\C$ is equal to the radial SLE$_2$ path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that $\p D$ is a $C^1$-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A\subset\p D$, is the chordal SLE$_8$ path in $\overline D$ joining the endpoints of A. A by-product of this result is that SLE$_8$ is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

The components of the Wired Spanning Forest are recurrent

We show that a.s. all of the connected components of the Wired Spanning Forest are recurrent, proving a conjecture of Benjamini, Lyons, Peres and Schramm. Our analysis relies on a simple martingale involving the effective conductance between the endpoints of an edge in a uniform spanning tree. We believe that this martingale is of independent interest and will find further applications in the study of uniform spanning trees and forests.

Competition for BLyS-mediated signaling through Bcmd/BR3 regulates peripheral B lymphocyte numbers

Striking cell losses occur during late B lymphocyte maturation, reflecting BcR-mediated selection coupled with requisites for viability promoting signals. How selection and survival cues are integrated remains unclear, but a key role for B lymphocyte stimulator (BLyS(TM); trademark of Human Genome Sciences, Inc.) is suggested by its marked effects on B cell numbers and autoantibody formation as well as the B lineage-specific expression of BLyS receptors. Our analyses of the B cell-deficient A/WySnJ mouse have established Bcmd as a gene controlling follicular B cell life span, and recent reports show Bcmd encodes a novel BLyS receptor. Here we show that A/WySnJ B cells are unresponsive to BLyS, affording interrogation of how Bcmd influences B cell homeostasis. Mixed marrow chimeras indicate A/WySnJ peripheral B cells compete poorly for peripheral survival. Moreover, in vivo BrdU labeling shows that (A/WySnJ x BALB/c)F(1) B cells have an intermediate but uniform life span, indicating viability requires continuous signaling via this pathway. Together, these findings establish the BLyS/Bcmd pathway as a dominant mediator of B cell survival, suggesting competition for BLyS/Bcmd signals regulates follicular B cell numbers.