Open quantitative problems for k-core percolation on the four-dimensional hypercubic lattice: surface scaling, nucleation kinetics, and the k=3 universality class

Parisi and Rizzo (Phys. Rev. E 78, 022101 (2008)) showed that k-core (bootstrap) percolation on the four-dimensional hypercubic lattice undergoes, for k=4, a discontinuous and strictly first-order transition at $p_c=0.6885(5)$ (finite correlation length, no precursor singularity), with a metastable percolating phase under periodic boundary conditions that decays by nucleation of low-density droplets, and, for k=3, a continuous transition whose critical behavior was left unmeasured. The analytical part of the open questions they raised — why the Bethe-lattice hybrid (mixed-order) singularity disappears in finite dimension — was later resolved by Rizzo's M-layer renormalization-group analysis (Phys. Rev. Lett. 122, 108301 (2019)), which showed that the mean-field hybrid critical point is washed out below the upper critical dimension. The surviving open core is quantitative: (i) the asymptotic form and possible divergence of the surface correction prefactor $c_1(p)$ as $p\to p_c^{+}$ for k=4; (ii) a quantitative characterization of the nucleation-driven decay of the metastable phase (nucleation rates, critical droplet sizes, and their dependence on $p$ and system size); and (iii) the critical exponents and universality class of the continuous k=3 transition in d=4, which a 2026 study confirms remains 'not firmly established'.

Background

k-core percolation, also called bootstrap percolation, was introduced by Chalupa, Leath and Reich as a model of dilute magnetic materials (J. Phys. C 12, L31 (1979)). Sites of a lattice are occupied independently with probability $p$; every site with fewer than $k$ occupied neighbors is then removed, and the culling is iterated until each remaining site has at least $k$ neighbors. The final set is the $k$-core. On the Bethe lattice — the infinite regular tree, where mean-field theory is exact — the $k$-core transition for $k\ge 3$ is hybrid (mixed-order): the order parameter (the $k$-core density $\rho$) jumps discontinuously at $p_c$ yet approaches its critical value with a square-root singularity, $\rho(p)\simeq\rho(p_c)+b(p-p_c)^{1/2}$, and the correlation length diverges. This mixed character made $k$-core percolation a paradigmatic mixed-order transition and a candidate lattice analogue of the jamming transition.

Whether the hybrid transition survives in finite dimension was a long-standing question. For $k>d$ on the $d$-dimensional hypercubic lattice the $k$-core is completely decimated for any $p<1$ in the infinite-size limit, so no transition exists. For $k\le d$, small self-sustained structures (such as full $d$-dimensional hypercubes) survive culling, and a percolation transition of the $k$-core can occur. In low dimensions the hybrid character is absent: for $k=2$ the transition is continuous and has the same critical point as ordinary percolation, and for $k=3$ in $d=3$ the transition is continuous with exponents consistent with ordinary percolation. A $1/d$ expansion by Harris and Schwarz (Phys. Rev. E 72, 046123 (2005)) found that to order $1/d^3$ the singularities of the order parameter and the susceptibility occur at the same value of $p$, suggesting that the hybrid transition might survive in high but finite dimensions.

Parisi and Rizzo (Phys. Rev. E 78, 022101 (2008)) studied numerically the four-dimensional hypercubic lattice, the first case with $3\le k\le d$ in which the hybrid transition was a live possibility. For $k=4$ they found a discontinuous transition at $p_c=0.6885(5)$ from a percolating high-density phase ($\rho_c^{+}=0.567(4)$) to a non-percolating low-density phase ($\rho_c^{-}=0.044(2)$), and established that it is strictly first order: the density shows no singular behavior before the jump and the correlation length $\xi$, extracted from the $k$-core two-point function, remains finite (estimated at about 2.5 lattice spacings at $p_c$, and never observed above roughly ten). In samples with periodic boundary conditions (PBC) the high-density phase can be followed below $p_c$ as a metastable state; it displays a pseudo-transition near $p\approx 0.686$ with the mean-field pseudo-exponent $1/2$ and an apparent correlation-length divergence, and it decays through the nucleation of droplets of the low-density phase seeded by rare density fluctuations. With empty (fully depleted) boundaries the finite-size density obeys $\rho_L(p)=\rho(p)-c_1(p)/L+O(L^{-2})$, where $c_1(p)=\int_0^{\infty}[\rho(p)-\rho(p,z)]\,dz$ is determined by the density profile at distance $z$ from the empty surface; $c_1(p)$ appeared to grow and diverge as $p\to p_c$, reflecting the increasingly deep penetration of the interface between the two phases. For $k=3$ in $d=4$ the transition was found to be continuous, but its critical behavior was not investigated.

The analytical part of this story was later settled. Rizzo (Phys. Rev. Lett. 122, 108301 (2019)) applied the M-layer construction — replicating the lattice $M$ times and randomly rewiring edges, so that $1/M$ plays the role of a bare coupling and a loop expansion around the Bethe solution becomes available — and showed that at all orders in this expansion $k$-core percolation in finite dimension is equivalent to a spinodal point — the endpoint of a metastable branch at which the free-energy barrier to nucleation vanishes — with quenched disorder, that is, to quadratic stochastic equations that are ill-defined beyond perturbation theory. Below the upper critical dimension $d_u=8$ (the dimension above which mean-field theory becomes exact) the mean-field hybrid critical point is therefore washed out as soon as one moves away from the Bethe solution; at most a pseudo-mean-field regime followed by an ordinary first-order transition can appear, which is exactly the phenomenology observed at $k=d=4$. This also consolidated the role of $k$-core percolation as a proxy for the avoided mode-coupling-theory singularity of supercooled liquids: mode-coupling theory predicts a sharp dynamical-arrest transition of glass-forming liquids that is not observed in practice, and $k$-core percolation reproduces the way this spurious singularity is rounded in finite dimension.

What survives from the 2008 open issues is quantitative. Three questions remain open: the scaling of the surface prefactor $c_1(p)$ as $p\to p_c^{+}$; a quantitative characterization of the nucleation-driven decay of the metastable PBC phase (rates, critical droplet sizes, dependence on $p$ and $L$); and the critical exponents and universality class of the continuous $k=3$ transition in $d=4$.

Problem Statement

The problem is to complete the quantitative characterization of the $k$-core percolation transitions on the four-dimensional hypercubic lattice $\mathbb{Z}^4$ in the regime $k\le d$ where the transition exists. Sites are occupied independently with probability $p$ and culled iteratively whenever they have fewer than $k$ occupied neighbors. Three linked questions, following from the numerical study of Parisi and Rizzo, remain:

1. Surface scaling at the $k=4$ transition. For $k=4$ the transition is discontinuous and strictly first order at $p_c=0.6885(5)$. In samples of linear size $L$ with empty boundaries the total density obeys $\rho_L(p)=\rho(p)-c_1(p)/L+O(L^{-2})$ with $c_1(p)=\int_0^{\infty}[\rho(p)-\rho(p,z)]\,dz$ fixed by the density profile at distance $z$ from an empty surface. Determine the asymptotic behavior of $c_1(p)$ as $p\to p_c^{+}$: does it diverge, and if so with what scaling law (exponent and, if possible, amplitude), or does it remain finite?

2. Nucleation of the metastable phase. Under periodic boundary conditions the high-density percolating phase persists below $p_c$ as a metastable state and decays (at $p\approx 0.6869$ for $L=320$, near a pseudo-transition at $p\approx 0.686$ with pseudo-exponent $1/2$) through the nucleation of droplets of the low-density phase seeded by density fluctuations; artificially inserted empty hypercubic regions shift the decay point to higher $p$. Provide a quantitative characterization of this nucleation process: the nucleation rate per unit volume, the critical droplet size, and their dependence on $p$ (the distance from $p_c$) and on $L$.

3. Critical behavior of the $k=3$ transition. For $k=3$ in $d=4$ the transition is continuous. Determine its critical exponents (order parameter, correlation length, susceptibility) and its universality class — in particular whether it coincides with that of ordinary four-dimensional percolation, as happens for $k=2$ and, numerically, for $k=3$ in $d=3$, or whether it defines a distinct universality class.

These three questions jointly constitute the still-missing quantitative description of the four-dimensional $k$-core transition landscape: the surface and finite-size physics of the $k=4$ discontinuous transition (questions 1 and 2) and the critical behavior of the $k=3$ continuous transition (question 3) on the same lattice. The originally co-equal fourth question — the analytical explanation of why the Bethe-lattice hybrid singularity disappears in finite dimension — has been resolved by Rizzo's M-layer analysis and is therefore not part of the problem.

References

1. G. Parisi and T. Rizzo, 'k-core percolation in four dimensions', Physical Review E 78, 022101 (2008). DOI: 10.1103/PhysRevE.78.022101, https://doi.org/10.1103/PhysRevE.78.022101; preprint arXiv:cond-mat/0609777, https://arxiv.org/abs/cond-mat/0609777 (the preprint carries the title 'On $k$-Core Percolation in Four Dimensions').
2. T. Rizzo, 'Fate of the Hybrid Transition of Bootstrap Percolation in Physical Dimension', Physical Review Letters 122, 108301 (2019). DOI: 10.1103/PhysRevLett.122.108301, https://doi.org/10.1103/PhysRevLett.122.108301; preprint arXiv:1807.08066, https://arxiv.org/abs/1807.08066 (the preprint carries the title 'The fate of the bootstrap percolation hybrid critical point in finite dimension').
3. A. B. Harris and J. M. Schwarz, '1/d expansion for k-core percolation', Physical Review E 72, 046123 (2005). DOI: 10.1103/PhysRevE.72.046123, https://doi.org/10.1103/PhysRevE.72.046123; preprint arXiv:cond-mat/0505329, https://arxiv.org/abs/cond-mat/0505329.
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7. J. Chalupa, P. L. Leath, and G. R. Reich, 'Bootstrap percolation on a Bethe lattice', Journal of Physics C: Solid State Physics 12, L31 (1979). DOI: 10.1088/0022-3719/12/1/008, https://doi.org/10.1088/0022-3719/12/1/008.
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