Universal inequalities for Dirichlet eigenvalues on discrete groups

Let \mathcal T be a (q+1) -regular tree and let \Gamma be a geometrically finite discrete subgroup of the group \operatorname{Aut}(\mathcal T) of automorphisms of \mathcal T . In this article, we prove an extreme value theorem on the distribution of geodesics in a non-compact quotient graph \Gamma\backslash\mathcal{T} . Main examples of such graphs are quotients of a Bruhat–Tits tree by non-cocompact discrete subgroups \Gamma of \operatorname{PGL}(2,\mathbf{K}) of a local field \mathbf{K} of positive characteristic. We investigate, for a given time T , the measure of the set of \Gamma -equivalent classes of geodesics with distance at most N(T) from a sufficiently large fixed compact subset D of \Gamma\backslash\mathcal{T} up to time T . We show that there exists a function N(T) such that for Bowen–Margulis measure \mu on the space \Gamma\backslash\mathcal{GT} of geodesics and the critical exponent \delta of \Gamma , \lim_{T\to\infty}\mu(\{[l]\in\Gamma\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N(T)+y\})=e^{-q^y/e^{2\delta y}}. In fact, we obtain a precise formula for N(T) : there exists a constant C depending on \Gamma and D such that N(T)=\log_{e^{2\delta/q}}\Big(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\Big).

The article continues the series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into the Laplacian part and degenerate part and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under tak- ing some central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelinization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph, nilpotent groups, Laplacian part of absolute.

Faber–Krahn type inequalities for trees

Elliptic gradient estimates for a nonlinear equation with Dirichlet boundary condition

Free products of completely positive maps and spectral sets

Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.

Let 5 be a discrete semigroup, P a probability measure on S and se S with lim sup"(P'",(j))1'" = l.We study limit theorems for the convolution powers P,n' ofP implied by the above property and further the class of all semigroups with this property.Theorem 3 relates this class of semigroups to left amenable semigroups. Introduction.Let S be a discrete semigroup and P a probability measure on S, that is a real valued function on S with P(s)~^.0 for all seS and 2sesF(i)=l.Kesten ([4], [5]) characterized amenable groups by means of the asymptotic behavior of convolution powers of symmetric probability measures defined on the group.A more precise information for the asymptotic behavior was obtained in [2] and [3] for symmetric probability measures on a discrete amenable group.In what follows we will derive similar theorems for probability measures on discrete semigroups.Let S be a discrete semigroup, P a probability measure on S. Then Supp P={s/P(s)>Q} denotes the support of P. To say Supp P generates S means: S= \J%=1 (SuppP)n.For probability measures P, Q on S define their convolution P * Q by P * Q(s) = 2 WOW SlS2=S (the summation is to be extended over all representations ois as a product of two elements sx, s2 of S).P * Q is again a probability measure and Supp P * g = (Supp P) (Supp Q).We often write P=P,P(") = P * /><-i>.Kesten obtained the following characterization of discrete amenable groups:Let G be a discrete group with unit element e, P a symmetric probability measure on G (P(g)=P(g~1) for all g in G) such that G is generated by

ABSTRACTIn this paper, we study the initial-boundary value problem of the generalized Burgers equation posed on a finite interval with non-homogeneous boundary conditions. The boundary conditions are given in a general form, which covers the usual Dirichlet, Neumann or Robin boundary conditions. For the generalized Burgers equation, we establish the local well-posedness for the weak solution in when the Sobolev index is negative. Besides, for the classical Burgers equation with Dirichlet boundary conditions, we obtain the global well-posedness.

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on G(n,d), a random d-regular graph on n vertices. It is well known that almost every such graph for d≥3 is an expander, and even essentially Ramanujan, implying a mixing time of O(logn). According to a conjecture of Peres, the simple random walk on G(n,d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6+o(1))log2n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on G(n,d). Namely, for any fixed d≥3, the simple random walk on G(n,d) whp has cutoff at (d/(d−2))logd−1n with window order logn. Surprisingly, the non-backtracking random walk on G(n,d) whp has cutoff already at logd−1n with constant window order. We further extend these results to G(n,d) for any d=no(1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers

We study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.

One-dimensional thermal wave transport in multilayered systems with an interface thermal resistance is studied under the framework of the Cattaneo–Vernotte hyperbolic heat conduction model, considering modulated heat excitation under Dirichlet and Neumann boundary conditions. For a single semi-infinite layer, analytical formulas useful in the measurement of its thermal relaxation time as well as additional thermal properties are presented. For a composite-layered system, in the thermally thin regime, with the Dirichlet boundary condition, the well known effective thermal resistance formula is obtained, while for the Neumann problem, only the heat capacity identity is found. In contrast, in the thermally thick case, an analytical expression for both Dirichlet and Neumann conditions is obtained for the effective thermal diffusivity of the whole system in terms of the thermal properties of the individual layers and their interface thermal resistance. The limits of applicability of this equation, in the thermally thick regime, are shown to provide useful and simple results in the characterization of layered systems and that they can be reduced to the results obtained using the Fourier approach. The role of the thermal relaxation time, the interface thermal resistance, and the implications of these results in the possibility of enhancement in heat transport are discussed.

This paper proposes a global approximation method to solve elliptic boundary value Poisson problems in arbitrary shaped 2‐D domains. Using transfinite interpolation, a symmetric finite element formulation is derived for degrees of freedom arranged mostly along the boundary of the domain. In cases where both Dirichlet and Neumann boundary conditions occur, the numerical solution is based on bivariate Coons interpolation using the boundary only. Furthermore, in case of only Dirichlet boundary conditions and no existing axes of symmetry, it is proposed to use at least one internal point and apply transfinite interpolation. The theory is sustained by five numerical examples applied to domains of square, circular and elliptic shape. Copyright © 2004 John Wiley &amp; Sons, Ltd.

On montre que si G est le produit libre de 2 groupes discrets G 1 , G 2 et si χ est une mesure sur G qui est une combinaison convexe de 2 mesures de probabilite χ 1 et χ 2 sur G 1 et G 2 , alors la resolvante (zSe-χ) −1 est proportionnelle au produit libre des resolvantes (λ i Se-χ i ) −1 , ou e est l'element identite et λ i depend analytiquement de z

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most n^{1/4 + o_n(1)} in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after n steps is n^{1/4 + o_n(1)}, as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. arXiv:1202.5454). More generally, we show that the simple random walks on a certain family of random planar maps in the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2)—including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance n^{1/d_gamma + o_n(1)} in n units of time, where d_gamma is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on gamma by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072). Since d_gamma > 2, this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into {mathbb {C}} wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.