Word maps and spectra of random graph lifts

Nati Linial, Doron Puder*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19-35), in almost every n-lift H of G, all "new" eigenvalues of H are ≤ O(λ1/21 ρ 1/2). Here we improve this bound to O(λ1/31 ρ2/3). It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19-35) that the statement holds with the bound ρ +o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995). For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d-regular graph has second eigenvalue O(d2/3). For the bouquet, Friedman (2008). has famously proved the (nearly?) optimal bound of 2 √ d-1 +o(1). Central to our work is a new analysis of formal words. Let w be a formal word in letters g± 1,...,g± 1. The word map associated with w maps the permutations σ1,...,σk ∈ Sn to the permutation obtained by replacing for each i, every occurrence of gi in w by σi. We investigate the random variable Xnw that counts the fixed points in this permutation when the σi are selected uniformly at random. The analysis of the expectation E(Xnw) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica (Nica, Random Struct Algorithms 5 (1994), 703-730), which determines, for every fixed w, the limit distribution (as n → ∞) of Xnw. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = ud for some word u.

Original languageAmerican English
Pages (from-to)100-135
Number of pages36
JournalRandom Structures and Algorithms
Issue number1
StatePublished - Aug 2010


  • Alon's conjecture
  • Graph lifts
  • Random graphs
  • Spectrum of graphs
  • Word map


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