A higher Boltzmann distribution

Abstract

We characterize the classical Boltzmann distribution as the unique solution to a combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex X and a choice of degree d dim X, we define by direct analogy a higher dimensional Boltzmann distribution q B as a certain real-valued cellular ðd À 1Þ-cycle. We then give an explicit formula for q B . We explain how these ideas relate to the Higher Kirchhoff Network Theorem of Catanzaro et al. (Homol Homotopy Appl 17:165-189, 2015). We also deduce an improved version of the Higher Matrix-Tree Theorems of Catanzaro et al. (Homol Homotopy Appl 17:165-189, 2015).

Harmonic Representatives in Homology over Arbitrary Fields

“…Thus dim(B k ) + dim(Z k ) = dim(C k ). Part 3: (5), (7), and (8) are equivalent. Equivalence of ( 5) and ( 7) follows from the general fact that, for linear maps f and g, ker(f • g) = ker(g) if and only if ker(f ) ∩ im(g) = 0.…”

“…Choosing f = ∂ * k+1 and g = ∂ k+1 gives the result. Lastly, items (7) and ( 8) are related by taking transposes. Indeed,…”

“…Spanning co-trees of the appropriate dimension always exist [7,Lemma 2.1]. Furthermore, the rational isomorphisms in their definition imply the relative integral homology group H k (X, L) is always finite for any k-dimensional spanning cotree L.…”

Correction to: A higher Boltzmann distribution

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The original publication of the article Catanzaro et al (2017) contained the following minor errors which we take this opportunity to correct:• In Remark 1.11: The statement, "Conditions (a)-(c) are equivalent to conditions (a'), (b) and (c)" should be replaced by "Conditions (a)-(c) imply conditions (a'), (b) and (c)" • In Remark 2.7:The sentence beginning with, "Let T d−1 (X ; Z) ⊂ …" until the end of the remark should be removed. • In the last paragraph of the proof of Theorem A:The text "…and φ L corresponds to A S .

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“…The original publication of the article Catanzaro et al (2017) contained the following minor errors which we take this opportunity to correct:…”