The asymptotic determinant of the discrete Laplacian

By Richard Kenyon. 2000

158 citations recorded by the source corpus.

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b9650ce8a07eccce5556c30bc50d469bc842f684

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Citations in the corpus

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori (2008, background): “Subsequent authors have studied asymptotics of the Laplacian on more general subgraphs of Z2, see, for example, BurtonPemantle [2] and Kenyon [17].”
Conformal invariance of planar loop-erased random walks and uniform spanning trees (2001, methodology): “Cauchy-Riemann equations, Richard Kenyon [ Ken00a , Ken00b] rigorously established the values of various critical exponents predicted for the LERW [GB90, Maj92, Dup92] in two dimensions.”
Random Complexes and l^2-Betti Numbers (2008, background): “We conjecture that the expected number of edges that belong to a cycle for the law P1 on an n × n square torus graph is asymptotic to Cn5/4 for some constant C; cf. Kenyon (2000).”
Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs (2011, methodology): “ to compute the probability of the connections 13|24 and 14|23. This computation can be used to compute the probability that an edge eis on the LERW from 1 to 2 (an equivalent calculation was done in [Ken00a]). Insert an extra edge e34 from 3 to 4; this “splits” the inner boundary into two (See Figu”
Conformally Invariant Processes in the Plane (2005, methodology): “The choice K = 2 was determined by a certain exponent for loop-erased walk that Kenyon had recently established rigorously [23] (also using a conformal invariance argument).”
Values of Brownian intersection exponents, II: Plane exponents (2000, background): “For instance, the definition of loop-erased random walks suggests that the number of vertices in the loop-erasure of an N2 step walk is roughly N2−ν(2,1) = N5/4 (loosely speaking, in order for one of the N2 steps of the simple random walk to remain in the loop-erasure, the future of the random walk ”
Conformally invariant scaling limits: an overview and a collection of problems (2006, methodology): “However, Kenyon’s estimate is much more precise; he shows that, in a specific domain, R3/4 times the probability is bounded away from zero and infinity, and in fact estimates the probability as f R−3/4 ( 1 + o(1) ) as R → ∞, where f is an explicit function of the positiong of the edge.”
The dimension of the SLE curves (2002, background): “walks [6], the expected number of steps grows in the appropriate way when the mesh of the lattice goes to zero, but its almost sure behaviour is not yet well-understood: For critical percolation, the up-to-constant estimate of the rst moment is missing, and for loop-erased random walks, it is the se”