For the ferromagnetic Ising model on uniformly sparse, locally tree-like random graphs and on the random Bethe tree with power-law degree tails of exponent τ>3\tau>3, Dommers, Giardinà and van der Hofstad (2014) rigorously identified the critical exponents β\boldsymbol{\beta}, δ\boldsymbol{\delta} and the high-temperature susceptibility exponent γ=1\boldsymbol{\gamma}=1, and proved only the lower bound γ′≥1\boldsymbol{\gamma}'\ge 1 for the low-temperature susceptibility. A detailed heuristic suggests χ(β,0+)≍(β−βc)−1\chi(\beta,0^+)\asymp(\beta-\beta_c)^{-1}, i.e. γ′=1\boldsymbol{\gamma}'=1 for all τ>3\tau>3, but the matching rigorous upper bound could not be established: moving an expectation inside an exponential of dependent cavity fields, and controlling constants sharply enough for an exact cancellation, remained open obstacles. The low-temperature specific-heat exponent α′\boldsymbol{\alpha}' (predicted α′=0\boldsymbol{\alpha}'=0 when E[K3]<∞\mathbb{E}[K^3]<\infty and α′=(τ−5)/(τ−3)\boldsymbol{\alpha}'=(\tau-5)/(\tau-3) for τ∈(3,5)\tau\in(3,5)) was left undetermined, as was the effect of slowly varying corrections to power-law degree tails for τ∈(3,5]\tau\in(3,5]. The problem is to prove or refute these predictions, completing the identification of the low-temperature critical behavior.

Background

The ferromagnetic Ising model on a finite graph G=(V,E)G=(V,E) with inverse temperature β≥0\beta\ge 0 and external field BB is the probability measure on spins {±1}V\{\pm1\}^{V} proportional to exp⁡{β∑(u,v)∈Eσuσv+B∑v∈Vσv}\exp\{\beta\sum_{(u,v)\in E}\sigma_u\sigma_v+B\sum_{v\in V}\sigma_v\}. Two infinite-family settings are studied. First, uniformly sparse, locally tree-like random graph sequences (Gn)(G_n) with asymptotic degree distribution P=(pk)k≥1P=(p_k)_{k\ge1}, such as configuration models (random graphs with a prescribed degree sequence) or generalized random graphs; 'locally tree-like' means the neighborhood of a typical vertex converges to a tree, and 'uniformly sparse' bounds the average degree. Second, the random Bethe tree T(D,K,∞)\mathcal{T}(D,K,\infty): an infinite rooted tree whose root has degree distributed as D∼PD\sim P, while every other vertex has a number of offspring distributed as the forward degree KK obtained from PP by size-biasing, P[K=k]=(k+1)pk+1/E[D]\mathbb{P}[K=k]=(k+1)p_{k+1}/\mathbb{E}[D], with ν=E[K]\nu=\mathbb{E}[K]. A locality theorem of Dembo and Montanari states that expectations of functions depending on finitely many vertices on GnG_n converge to the corresponding values on T(D,K,∞)\mathcal{T}(D,K,\infty). The model undergoes a ferromagnetic phase transition at βc=atanh⁡(1/ν)\beta_c=\operatorname{atanh}(1/\nu) (with βc=0\beta_c=0 when ν=∞\nu=\infty, i.e. τ∈(2,3]\tau\in(2,3]). The analysis rests on the cavity (belief-propagation) recursion h=dB+∑i=1Kξ(hi)h\stackrel{d}{=}B+\sum_{i=1}^{K}\xi(h_i) for the effective field acting on a vertex, where ξ(h)=atanh⁡(β^tanh⁡h)\xi(h)=\operatorname{atanh}(\hat\beta\tanh h), β^=tanh⁡β\hat\beta=\tanh\beta, and the hih_i are independent copies; it has a unique fixed point for every β≥0\beta\ge0 and B>0B>0. On graphs the susceptibility is χn(β,B)=1n∑i,j(⟨σiσj⟩−⟨σi⟩⟨σj⟩)\chi_n(\beta,B)=\frac1n\sum_{i,j}(\langle\sigma_i\sigma_j\rangle-\langle\sigma_i\rangle\langle\sigma_j\rangle), equal to the second BB-derivative of the pressure per particle, and χ(β,B)\chi(\beta,B) is its thermodynamic limit; the specific heat is the second β\beta-derivative of the pressure, i.e. the variance of the energy per site. Writing f≍gf\asymp g when f/gf/g is bounded away from 00 and ∞\infty, the critical exponents are defined by M(β,0+)≍(β−βc)βM(\beta,0^+)\asymp(\beta-\beta_c)^{\boldsymbol{\beta}} as β↘βc\beta\searrow\beta_c (MM the magnetization), M(βc,B)≍B1/δM(\beta_c,B)\asymp B^{1/\boldsymbol{\delta}}, χ(β,0+)≍(βc−β)−γ\chi(\beta,0^+)\asymp(\beta_c-\beta)^{-\boldsymbol{\gamma}} as β↗βc\beta\nearrow\beta_c, and χ(β,0+)≍(β−βc)−γ′\chi(\beta,0^+)\asymp(\beta-\beta_c)^{-\boldsymbol{\gamma}'} as β↘βc\beta\searrow\beta_c; the specific-heat exponent α′\boldsymbol{\alpha}' describes the singularity of the specific heat as β↘βc\beta\searrow\beta_c, negative values corresponding to a finite cusp rather than a divergence. For power-law degree tails ∑ℓ≥kpℓ∼c k−(τ−1)\sum_{\ell\ge k}p_\ell\sim c\,k^{-(\tau-1)}, physics predictions of Dorogovtsev, Goltsev and Mendes, and of Leone, Vázquez, Vespignani and Zecchina, state: for τ>5\tau>5 (equivalently E[K3]<∞\mathbb{E}[K^3]<\infty) the mean-field values β=1/2\boldsymbol{\beta}=1/2, δ=3\boldsymbol{\delta}=3, γ=1\boldsymbol{\gamma}=1, α′=0\boldsymbol{\alpha}'=0; for 3<τ<53<\tau<5 the anomalous values β=1/(τ−3)\boldsymbol{\beta}=1/(\tau-3), δ=τ−2\boldsymbol{\delta}=\tau-2, γ=1\boldsymbol{\gamma}=1, α′=(τ−5)/(τ−3)\boldsymbol{\alpha}'=(\tau-5)/(\tau-3); with logarithmic corrections at the boundary τ=5\tau=5. Dommers, Giardinà and van der Hofstad proved all the unprimed predictions in both settings, the exact subcritical susceptibility χ(β,0+)=1+E[D]β^/(1−νβ^)\chi(\beta,0^+)=1+\mathbb{E}[D]\hat\beta/(1-\nu\hat\beta) for β<βc\beta<\beta_c, and the lower bound γ′≥1\boldsymbol{\gamma}'\ge 1; however they could not prove that γ′\boldsymbol{\gamma}' exists or equals 11. Their heuristic upper-bound argument expands χ\chi as a sum over tree depths ℓ\ell of terms (β^ν)ℓ(\hat\beta\nu)^\ell damped by the exponential of accumulated fluctuations of the cavity fields hi⋆h_i^\star (the size-biased fields along a ray, satisfying an approximate fixed-point recursion); since β^ν>1\hat\beta\nu>1 for β>βc\beta>\beta_c, the damping must be extracted precisely. Making this rigorous fails at two identified points: (a) the expectation is moved inside the exponential although the fields hi⋆h_i^\star along a path are dependent, and (b) a positive and a negative term cancel exactly in the final constant, requiring sharper control of constants than the available moment estimates provide. The heuristic also predicts the sharp amplitude lim⁡β↘βc(β^ν−1)χ(β,0+)=E[D]/(2ν)\lim_{\beta\searrow\beta_c}(\hat\beta\nu-1)\chi(\beta,0^+)=\mathbb{E}[D]/(2\nu), half the subcritical amplitude, matching the Curie-Weiss model. The specific-heat exponent α′\boldsymbol{\alpha}' is a second-order quantity that is not a local function of the graph, and the authors state they cannot relate the specific heat on the random graph to that on the random Bethe tree. Finally, they expect that slowly varying corrections to the power-law tail, ∑ℓ≥kpℓ≍L(k)k−(τ−1)\sum_{\ell\ge k}p_\ell\asymp L(k)k^{-(\tau-1)} with LL slowly varying (i.e. L(ck)/L(k)→1L(ck)/L(k)\to1 as k→∞k\to\infty for every fixed c>0c>0), modify the critical behavior for τ∈(3,5]\tau\in(3,5] (for instance L(k)=(log⁡k)−2L(k)=(\log k)^{-2} at τ=5\tau=5 removes the logarithmic corrections), but leave this open.

Problem Statement

Let (Gn)n≥1(G_n)_{n\ge1} be a uniformly sparse, locally tree-like sequence of random graphs with asymptotic degree distribution P=(pk)k≥1P=(p_k)_{k\ge1} (e.g. a configuration model or generalized random graph), and let T(D,K,∞)\mathcal{T}(D,K,\infty) be the corresponding random Bethe tree with forward-degree variable KK and ν=E[K]\nu=\mathbb{E}[K]. For items 1–2, assume either E[K3]<∞\mathbb{E}[K^3]<\infty or a pure power-law tail ∑ℓ≥kpℓ∼c k−(τ−1)\sum_{\ell\ge k}p_\ell\sim c\,k^{-(\tau-1)} with τ∈(3,5]\tau\in(3,5]; item 3 instead considers the broader class of degree tails ∑ℓ≥kpℓ≍L(k) k−(τ−1)\sum_{\ell\ge k}p_\ell\asymp L(k)\,k^{-(\tau-1)} with LL slowly varying and τ∈(3,5]\tau\in(3,5]. For the ferromagnetic Ising model on GnG_n and on T(D,K,∞)\mathcal{T}(D,K,\infty), with susceptibility χ(β,B)\chi(\beta,B) (the thermodynamic limit of 1n∑i,j(⟨σiσj⟩−⟨σi⟩⟨σj⟩)\frac1n\sum_{i,j}(\langle\sigma_i\sigma_j\rangle-\langle\sigma_i\rangle\langle\sigma_j\rangle) on the graphs, the BB-derivative of the root magnetization on the tree) and critical point βc=atanh⁡(1/ν)\beta_c=\operatorname{atanh}(1/\nu):

  1. (Low-temperature susceptibility exponent) Prove that the critical exponent γ′\boldsymbol{\gamma}' exists and equals 11, i.e. χ(β,0+)≍(β−βc)−1\chi(\beta,0^+)\asymp(\beta-\beta_c)^{-1} as β↘βc\beta\searrow\beta_c, in both settings and for all admissible degree distributions above — the lower bound γ′≥1\boldsymbol{\gamma}'\ge 1 is known, and the open part is the matching upper bound; the source's heuristic argument for this upper bound is not rigorous, because the exchange of expectation and exponential in the cavity-field expansion of χ\chi is unjustified for the dependent hi⋆h_i^\star fields along a ray, and because the sharp constants needed for the cancellation on which that heuristic relies are unavailable; or rigorously establish that γ′≠1\boldsymbol{\gamma}'\ne 1 for some admissible degree distribution.

  2. (Specific-heat exponent) Determine the specific-heat exponent α′\boldsymbol{\alpha}' in the low-temperature phase — the order of the singularity of the second β\beta-derivative of the pressure per particle as β↘βc\beta\searrow\beta_c — proving or refuting the predictions α′=0\boldsymbol{\alpha}'=0 when E[K3]<∞\mathbb{E}[K^3]<\infty and α′=(τ−5)/(τ−3)\boldsymbol{\alpha}'=(\tau-5)/(\tau-3) when τ∈(3,5)\tau\in(3,5); the available locality arguments do not relate the specific heat on the random graph to that on the random Bethe tree.

  3. (Slowly varying tails) For degree tails ∑ℓ≥kpℓ≍L(k) k−(τ−1)\sum_{\ell\ge k}p_\ell\asymp L(k)\,k^{-(\tau-1)} with LL slowly varying (i.e. L(ck)/L(k)→1L(ck)/L(k)\to1 as k→∞k\to\infty for every fixed c>0c>0) and τ∈(3,5]\tau\in(3,5], identify how LL modifies the critical behavior, including the asymptotics in items 1–2 and the appearance or disappearance of logarithmic corrections (e.g. L(k)=(log⁡k)−2L(k)=(\log k)^{-2} at τ=5\tau=5 is expected to remove them).

The common scientific objective is to complete the identification of the low-temperature critical behavior of the Ising model on locally tree-like random graphs predicted by Dorogovtsev–Goltsev–Mendes and by Leone et al.: item 1 is the primary and sharpest target (its identified technical core is the constant and dependency control), item 2 is its second-order companion on the same phase transition, and item 3 is the same question under the general tail assumption. Partial rigorous resolution of one component is substantive progress toward this objective; a complete answer establishes, proves or refutes the predicted values for the stated classes, without restricting to a narrower model class (such as regular degrees).

References

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