---
title: Multicritical Contact Process
url: https://www.emergentmind.com/topics/multicritical-contact-process
type: topic
---

# Multicritical Contact Process

Searching arXiv for recent and foundational papers on multicritical contact-process variants, disorder-driven multicriticality, and boundary/defect-induced mixed-order behavior.
arXiv search 1: query = "all:multicritical contact process"
The multicritical contact process denotes a family of absorbing-state stochastic lattice models in which criticality is governed by more than a single scalar control parameter or by more than one relevant critical mechanism. In the classical one-parameter contact process, survival is controlled by a unique threshold and the transition is in the directed percolation (DP) universality class. By contrast, multicritical variants replace the single threshold by a critical curve, a critical surface, or a mixed-order local transition, as happens in boundary-enhanced one-dimensional models, disordered systems where percolation and infection disorder compete, and multiscale or quantum extensions [2312.02059] [2508.20895] [1101.1165]. A distinct mechanical literature uses “contact” in the sense of force-bearing contact networks in jamming rather than the stochastic contact process; that usage concerns rigidity percolation, not epidemic-like absorbing-state dynamics [1809.09631].

## 1. Ordinary contact process and the meaning of multicriticality

The ordinary contact process is a continuous-time Markov process on a lattice or graph whose states are active or inactive, with spontaneous healing at rate \(1\) and nearest-neighbor infection at rate \(\lambda\). It has an absorbing state and a single critical value \(\lambda_c\), below which activity dies out and above which an active phase survives. In one and two dimensions, and more generally in short-range single-order-parameter settings without extra symmetries or conservation laws, this transition is the standard DP transition [1802.00037] [1902.04515].

In the contact-process literature, “multicritical” is used in several related senses. One is genuinely multi-parameter criticality, where survival depends on two infection parameters and the phase boundary is a critical curve rather than a point. Another is competition between different relevant disorder mechanisms, such as geometric dilution and infection-rate disorder, producing a multicritical point where distinct critical lines meet. A third is multiscale criticality, where different subsystems or observables are controlled by different effective scales and exponent sets, as in hybrid lattice–network models [2312.02059] [2508.20895] [1101.1165].

The same term also appears in settings where the bulk transition remains DP-like but local critical behavior becomes mixed-order because of geometry or inhomogeneity. Multiple junctions of semi-infinite chains, marginal extended defects, and boundary-driven asymmetric dynamics all produce local phenomena that are not captured by homogeneous DP scaling, even though the underlying model is still a contact process [1612.02999] [1711.03495] [1007.2773].

## 2. Two-parameter boundary enhancement and the critical curve

A direct realization of a multicritical contact process is the one-dimensional contact process with enhancements studied on \(\mathbb{Z}\), where the infected set \(\eta_t \subseteq \mathbb{Z}\) evolves with two infection parameters: an interior rate \(\lambda_i\) and a boundary rate \(\lambda_e\). If \(B\subseteq \mathbb{Z}\) is the current infected set, then each infected site recovers at rate \(1\), while infection of a vacant nearest neighbor occurs at rate \(\lambda_i\) when the source is not the leftmost or rightmost infected site of a finite cluster, and at rate \(\lambda_e\) when the source is the leftmost or rightmost infected site. This makes the enhancement dynamic and geometry-dependent rather than localized on a fixed set of edges [2312.02059].

The survival probability from a single infected site is
\[
\theta(\lambda_i,\lambda_e)=\mathbb{P}\big(\eta_t^{\{0\}}\neq\emptyset\ \forall t\ge 0\big).
\]
The standard critical value is
\[
\lambda_c=\inf\{\lambda:\theta(\lambda,\lambda)>0\},
\]
while the two critical functions are
\[
\lambda_*^i(\lambda_e)=\inf\{\lambda:\theta(\lambda,\lambda_e)>0\},
\qquad
\lambda_*^e(\lambda_i)=\inf\{\lambda:\theta(\lambda_i,\lambda)>0\}.
\]
These functions define a critical curve in the \((\lambda_i,\lambda_e)\)-plane separating extinction from survival. The principal structural result is that \(\lambda_*^i(\lambda_e)\) is strictly decreasing for \(\lambda_e\in(1,\lambda_c]\), \(\lambda_*^e(\lambda_i)\) is strictly decreasing on \([\lambda_c,\infty)\), both are continuous on their domains, and they are inverse to one another in the sense that \(\lambda_*^e(a)=b\) iff \(\lambda_*^i(b)=a\) [2312.02059].

This model exhibits a genuine multicritical point at \((\lambda_i,\lambda_e)=(\lambda_c,\lambda_c)\). At that point the standard critical contact process dies out, but either coordinate direction immediately produces survival: if \(\lambda_i=\lambda_c\) and \(\lambda_e>\lambda_c\), then \(\theta(\lambda_c,\lambda_e)>0\); if \(\lambda_e=\lambda_c\) and \(\lambda_i>\lambda_c\), then \(\theta(\lambda_i,\lambda_c)>0\). The effect is asymmetric in mechanism but symmetric in phase-diagram structure: boundary enhancement alone can rescue a bulk-critical process, and bulk enhancement alone can rescue a boundary-critical process [2312.02059].

A second organizing quantity is the asymptotic speed of the rightmost infected site when starting from a semi-infinite initial condition,
\[
\alpha_t(\lambda_i,\lambda_e)=\mathbb{E}[R\eta_t^-],\qquad
\alpha(\lambda_i,\lambda_e)=\inf_{t>1}\frac{\alpha_t(\lambda_i,\lambda_e)}{t}.
\]
In the attractive regime \(\lambda_e\le \lambda_i\), the limit \(R\eta_t^-/t\to \alpha(\lambda_i,\lambda_e)\) exists almost surely. On the critical curve, \(\alpha=0\); in the supercritical region, \(\alpha>0\). The strict increase of \(\alpha\) with either parameter gives a sharp dynamical counterpart to the monotone critical curve and makes the multicritical transition visible as a change from zero to positive linear growth speed [2312.02059].

## 3. Disorder-driven multicritical points and activated scaling

A different notion of multicriticality arises when distinct disorder mechanisms compete. In the multicritical infection-spreading problem studied on bond-diluted hypercubic lattices, the control parameters are the bond-dilution probability \(p\) and the infection rate \(\lambda\). The phase diagram contains a percolation transition line at \(p=p_c\) and a generic disordered absorbing-state transition line for \(p<p_c\); these meet at a multicritical point \((p_c,\lambda_{\mathrm{MCP}})\). At this point the critical behavior is universal and displays activated, ultra-slow scaling, with exponents consistent with strong-disorder renormalization group predictions for the multicritical random transverse-field Ising model [2508.20895].

At the multicritical point, the disorder-averaged observables obey
\[
\rho(t),\,P(t)\sim [\ln(t/t_0)]^{-\bar\delta},\qquad
N(t)\sim [\ln(t/t_0)]^{\bar\Theta},\qquad
R(t)\sim [\ln(t/t_0)]^{1/\psi}.
\]
In two dimensions the best estimate is \(\lambda_c^{(2)}=3.5588(13)\), and in three dimensions \(\lambda_c^{(3)}=5.59(1)\). The exponent combinations extracted from \(R(P)\), \(N(P)\), and \(N(R)\) agree with the SDRG multicritical Ising values in both dimensions, which supports the statement that the multicritical contact process and the multicritical quantum Ising model belong to the same universality class [2508.20895].

Related disordered contact processes supply the broader scaling framework. On a square lattice with quenched site dilution, the critical dynamics are governed by an infinite-randomness fixed point with activated scaling rather than DP power laws. The dynamic critical behavior is compatible with the universality class of the random transverse-field Ising model, and the phase diagram contains active percolating, inactive percolating, and nonpercolating phases [1703.09261]. This does not by itself define a multicritical point, but it shows how quenched geometry and absorbing-state dynamics can combine to produce non-DP universality.

When spatial and temporal disorder are both present but decouple as \(\lambda(x,t)=\lambda_0 f(x)g(t)\), generalized Harris criteria imply that clean DP is unstable to either disorder type, the infinite-randomness fixed point is stable against weak temporal disorder, and the infinite-noise fixed point is stable against weak spatial disorder. Large-scale simulations then suggest a multicritical regime when the two disorder strengths are comparable, together with modified Griffiths singularities: power-law Griffiths behavior can cross over to stretched-exponential decay or lifetimes when the second disorder source becomes strong enough [2207.11798]. In this usage, multicriticality refers to the meeting of two disorder-dominated universality classes rather than to a simple two-coupling phase boundary.

## 4. Boundary, junction, and defect-induced mixed-order criticality

Multicritical behavior can also be local rather than bulk. For the contact process near a multiple junction of \(M\) semi-infinite chains, the local order parameter at the junction is continuous for \(M=1\) and \(M=2\), but becomes discontinuous for \(M>2\). At the same time the temporal correlation length diverges algebraically on approaching the critical point, with different exponents on the two sides of the transition: on the active side the estimate is compatible with the bulk value, whereas on the inactive side it exceeds the bulk value and increases with \(M\). This is a mixed-order local transition, and quenched spatial disorder restores continuity, consistent with earlier RG results [1612.02999].

A closely related phenomenon occurs near an extended surface defect with
\[
\lambda(l)-\lambda(\infty)=A\,l^{-s},
\]
on a semi-infinite one-dimensional chain. The marginal case is \(s=1/\nu_\perp\). For \(A<A_c\), the surface transition is continuous and the surface order-parameter exponent varies continuously with \(A\). For \(A>A_c\), the surface order parameter is discontinuous at the bulk critical point while the temporal correlation length diverges algebraically, again with different exponents on the two sides. The paper estimates \(A_c\approx 3.25(10)\), finds
\[
p(A)\sim (A-A_c)^{\beta_{tc}},\qquad \beta_{tc}=0.40(4),
\]
for the onset of the critical surface order parameter in the mixed-order regime, and observes logarithmic decay \(P(t)\sim [\ln t]^{-\gamma}\) with \(\gamma\approx 0.53\) at the tricritical point [1711.03495].

These two settings are linked by the same scaling mechanism. In the active phase the local control variable is an irrelevant perturbation in the ordinary sense, while in the inactive phase it becomes a dangerous irrelevant variable. That distinction generates the exponent asymmetry and explains why a discontinuous local order parameter can coexist with a diverging temporal correlation length [1612.02999] [1711.03495].

A separate but related boundary phenomenon occurs in the boundary-driven asymmetric contact process with an active wall. There the bulk order parameter still undergoes a continuous DP transition, but the apparent velocity of the travelling activity front jumps discontinuously at the critical point. A modified Fisher equation shows that the intrinsic front velocity changes smoothly, while the observable discontinuity arises from the change of the envelope from localized to propagating [1007.2773]. This is not a multicritical point in the strict RG sense, but it illustrates how boundary drive and bias can superpose first-order-like dynamical singularities on top of ordinary absorbing-state criticality.

## 5. Multicomponent, multiscale, and age-structured classical variants

Multicritical contact-process behavior is also realized by adding internal states, species, or network scales. In the symbiotic two-species contact process, each site can be vacant or occupied by \(A\), \(B\), or \(AB\), and death at doubly occupied sites is reduced to \(\mu<1\). Mean-field theory predicts a discontinuous transition for \(\mu<1/2\), but Monte Carlo simulations, field theory, and later rigorous work all indicate that the phase transition is continuous and in the DP universality class. At the same time the critical creation rate \(\lambda_c(\mu)\) decreases with \(\mu\), and the rigorous asymptotics give \(\lambda_c(\mu)\) of order \(\sqrt{\mu}\) as \(\mu\to 0\) [1205.5974] [1904.02213]. The controversy here is not whether the phase boundary moves—it does—but whether symbiosis produces a true discontinuous or multicritical bulk transition; the available evidence in the cited works supports continuity.

The contact process with aging enlarges the state space from \(\{0,1\}^{\mathbb{Z}^d}\) to \(N^{\mathbb{Z}^d}\), where each living particle carries an integer age and the birth rates form a nondecreasing sequence \(\Lambda=(\lambda_i)\) with limiting value \(\lambda_\infty\), while maturation occurs at rate \(\gamma\). This model embeds the two-stage contact process, admits a coupling to supercritical oriented percolation, and satisfies a shape theorem conditioned on survival. The extra age variable does not by itself define a multicritical point in the paper, but it turns the usual single threshold into a higher-dimensional parameter region and provides a rigorous framework for multi-stage generalizations [1405.6153].

A distinct multiscale construction arises when one-dimensional chains are coupled through a Barabási–Albert scale-free network and infected individuals travel between chains at rate \(\alpha\). The model has two size variables, chain length \(L\) and number of cities \(N\), and a generalized finite-size scaling Ansatz
\[
\overline{\rho}_s(\Delta,L,N)=L^{-\beta/\nu_\perp}N^{-\beta'/\nu_\perp'}\,
\mathcal{F}\!\left(\Delta L^{1/\nu_\perp}N^{1/\nu_\perp'}\right).
\]
It exhibits a finite epidemic threshold, an epidemic mean lifetime diverging exponentially in the subcritical phase, power-law divergence of outbreak duration in quasistationary analysis, and a new universality class associated with the hybrid local–global architecture [1101.1165]. Here multicriticality refers to the coexistence of two relevant spatial scales and two dynamical scales rather than to a single isolated point.

## 6. Quantum, quasiperiodic, and universality issues

The quantum contact process replaces classical branching by coherent nearest-neighbor-conditioned flips of a spin-\(\tfrac12\) chain with Hamiltonian
\[
H=\Omega\sum_{k=1}^{L-1}\bigl(\sigma_1^{(k)}n^{(k+1)}+n^{(k)}\sigma_1^{(k+1)}\bigr),
\]
combined with incoherent decay through a Lindblad dissipator with jump operators \(L_k=\sqrt{\gamma}\,\sigma_-^{(k)}\). Real-time iTEBD simulations in one dimension provide strong evidence for a continuous absorbing-state transition at \(\Omega_c\approx 6\), but with exponents that differ from both one- and two-dimensional DP and are strikingly close to those of a tricritical point in a two-dimensional quantum-plus-classical branching model. The paper therefore connects the one-dimensional quantum contact process to tricritical behavior without claiming a fully established new universality class [1902.04515]. This is a clear example of multicriticality entering through coherent dynamics rather than through extra classical infection parameters.

Quasiperiodic modulation offers another route. In the contact process on generalized Fibonacci chains defined by the inflation rules \(A\to AB^k\), \(B\to A\), the wandering exponent \(\omega\) determines whether the modulation is irrelevant or relevant. For \(k=1,2\), the transition remains in the clean DP class; for \(k\ge 3\), the system flows to an unconventional infinite-modulation critical point with activated scaling,
\[
\ln \xi_\parallel \sim \xi_\perp^\psi,\qquad \psi=\omega,
\]
and with pronounced double-log periodic oscillations in observables such as the survival probability and the size of the active cloud [1310.2976]. This is not multicriticality in the two-parameter phase-diagram sense, but it is a sharp example of how deterministic inhomogeneity can move the contact process away from DP to a different strong-modulation critical structure.

Against this background, studies of ordinary two-dimensional contact-process variants provide a useful reference point. The standard continuous-time process, a discrete-time variant, and a threshold-contagion version all share the same ageing exponents, global and local two-time scaling functions, and critical behavior in the DP class, despite differing microscopic rules [1802.00037]. That result matters because it isolates what multicritical variants must actually change: not merely microscopic update details, but the number of relevant couplings, the geometry of the substrate, the disorder structure, or the dynamical algebra itself.

Taken together, these developments support a broad but precise encyclopedia definition. A multicritical contact process is not a single model but a class of contact-process extensions in which criticality is organized by more than one relevant field or mechanism. In the most direct cases this produces a strictly monotone critical curve in a two-parameter infection space [2312.02059]; in disordered settings it produces a multicritical point with activated scaling and random-Ising universality [2508.20895]; and in boundary, defect, multiscale, symbiotic, aging, quantum, and quasiperiodic variants it produces a hierarchy of local, mixed-order, crossover, or tricritical phenomena that extend the standard DP paradigm without eliminating its role as the principal reference theory.

Source: https://www.emergentmind.com/topics/multicritical-contact-process