scholarly journals Coverings, Laplacians, and Heat Kernels of Directed Graphs

10.37236/120 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Clara E. Brasseur ◽  
Ryan E. Grady ◽  
Stratos Prassidis
Keyword(s):  
Random Walk ◽  
Heat Kernel ◽  
Directed Graphs ◽  
Heat Kernels ◽  
Total Space ◽  
Regular Tree ◽  
Combinatorial Laplacian

Combinatorial covers of graphs were defined by Chung and Yau. Their main feature is that the spectra of the Combinatorial Laplacian of the base and the total space are related. We extend their definition to directed graphs. As an application, we compute the spectrum of the Combinatorial Laplacian of the homesick random walk $RW_{\mu}$ on the line. Using this calculation, we show that the heat kernel on the weighted line can be computed from the heat kernel of '$(1 + 1/\mu)$-regular' tree.

scholarly journals Coverings, Heat Kernels and Spanning Trees

10.37236/1444 ◽  
1998 ◽  
Vol 6 (1) ◽  
Author(s):  
Fan Chung ◽  
S.-T. Yau
Keyword(s):  
Heat Kernel ◽  
Regular Graph ◽  
Spanning Trees ◽  
Constant Factor ◽  
Heat Kernels ◽  
Regular Graphs ◽  
Regular Tree

We consider a graph $G$ and a covering $\tilde{G}$ of $G$ and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite $k$-regular tree and we examine the heat kernels for general $k$-regular graphs. In particular, we show that a $k$-regular graph on $n$ vertices has at most $$ (1+o(1)) {{2\log n}\over {kn \log k}} \left( {{ (k-1)^{k-1}}\over {(k^2-2k)^{k/2-1}}}\right)^n $$ spanning trees, which is best possible within a constant factor.

scholarly journals Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

2010 ◽  
Vol 198 ◽  
pp. 121-172
Author(s):  
Gautam Chinta ◽  
Jay Jorgenson ◽  
Anders Karlsson
Keyword(s):  
Heat Kernel ◽  
Modular Forms ◽  
Spanning Trees ◽  
Zeta Functions ◽  
Constant Term ◽  
Finite Graph ◽  
Heat Kernels ◽  
Careful Study ◽  
Trigonometric Functions ◽  
Combinatorial Laplacian

AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called thecomplexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study ofI-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

scholarly journals Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

2010 ◽  
Vol 198 ◽  
pp. 121-172 ◽  
Author(s):  
Gautam Chinta ◽  
Jay Jorgenson ◽  
Anders Karlsson
Keyword(s):  
Heat Kernel ◽  
Modular Forms ◽  
Spanning Trees ◽  
Zeta Functions ◽  
Constant Term ◽  
Finite Graph ◽  
Heat Kernels ◽  
Careful Study ◽  
Trigonometric Functions ◽  
Combinatorial Laplacian

AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

A proof of the main identity

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Heat Kernel ◽  
Heat Kernels

This chapter proves the formula that was stated in Chapter 6. It first states various estimates on the hypoelliptic heat kernels, which are valid for b ≥ 1 and then makes a natural rescaling on the coordinates parametrizing ̂X. Next, the chapter introduces a conjugation on the Clifford variables and shows that the norm of the term defining the conjugation can be adequately controlled. The chapter then introduces a conjugate ℒA,bX of another ℒA,bX and its associated heat kernel. Afterward, the chapter obtains the limit as b → +∞ of the rescaled heat kernel, thus establishing the formula in Chapter 6. After further computations, this chapter states a result on convergence of heat kernels.

Heat Kernels of Lorentz Cones

1999 ◽  
Vol 42 (2) ◽  
pp. 169-173 ◽  
Author(s):  
Hongming Ding
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Lower Bounds ◽  
Heat Kernels ◽  
Upper And Lower Bounds ◽  
Symmetric Cones ◽  
Lorentz Cone

AbstractWe obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time t and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

Poisson’s integral formula via heat kernels

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 59-64
Author(s):  
Yoichi Miyazaki
Keyword(s):  
Fourier Transform ◽  
Heat Kernel ◽  
Half Space ◽  
Unbounded Domain ◽  
Harmonic Functions ◽  
Integral Formula ◽  
Heat Kernels ◽  
Transform Method ◽  
Fourier Transform Method ◽  
The Fourier Transform

Abstract We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

scholarly journals Exponential coordinates and regularity of groupoid heat kernels

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Bing So
Keyword(s):  
Differential Operator ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Pseudo Differential Operator

AbstractWe prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.

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