In this short introduction I present my primary field of interest through references of works I have been involved in. Of course these references do not form a complete bibliography of the field.

Continuous phase transitions in equilibrium statistical systems has been found to belong universality classes which are independent of the details of the interactions of finite range. Renormalization group (RG) method investigations has shown that they depend on the dimension and the symmetries of systems. Exact analytical treatment is not possible for most of the model systems investigated but approximate RG studies, numerical simulations have been used to explore the critical behavior (see Refs. [1,2,4,5]).

In non-equilibrium systems phase transition may occur as low as in one dimension since there are less interacting neighbors of variables and fluctuations can destroy ordered states. One might expect easier treatment of low dimensional models, but in fact it is hard to describe fluctuation dominated systems. Most methods start from fluctuation free, mean-field solutions (that is valid above a critical dimension) and add fluctuations perturbatively. The understanding of non-equilibrium phase transitions is incomplete as compared to equilibrium ones [42].

The dynamical critical behavior has been found to depend on **initial
conditions ** in many cases
(see Refs. [19,20,22,24,25,28,27]).
This means that it takes infinitely long time until the natural steady
state correlations build up. This can happen when we introduce long-range
initial correlations [19,24]
or when we couple to another system that is either slowly varying [20,30]
or creates blockades by **hard-core interactions
**[22,25,28,27]
between particles.

In the latter case the static exponents are effected as well and
new, robust universality classes emerge [28,27,34]
in one dimension that are insensitive to conservation laws that has been
found to be relevant in other non-equilibrium systems.

Such a conservation law is the parity of the number of particles that
is responsible for the **PC **class in
one-dimensional reaction diffusion models [13,15,16,17,18,24,45].

This kind of branching and annihilating random walks with two offspring's
(BAW2) arise in certain Ising systems (NEKIM) as kinks between ordered
domains as well, but here the **Z2 symmetry**
between the ordered, absorbing states is necessary condition too
[15,
17,
24,
35,
40]
for the appearance of the PC class.
At this critical behavior besides scaling a more general
**local scale invariance (LSI) **
of two-point, ageing functions seems
also be present asymptotically [48].
By destroying that symmetry either
by an external field [15]
or by fluctuations [17]
one obtains another robust class, the directed percolation class (DP).
The main representative model of this class is the branching and annihilating
random walk with one offspring (BAW1):

The DP class has been identified in many systems and according to
the hypothesis of Janssen and Grassberger all continuous phase transitions
to a single absorbing state in homogeneous systems with short ranged interactions
belong to this class provided there is no additional symmetry and quenched
randomness present. Unfortunately there have not been found too many experimental
realizations of this process yet, because any small imperfection causes
relevant perturbation and changes the scaling behavior.

I have been involved in the confirmation of this DP hypothesis for cellular automata type systems using simulations and cluster mean-field approximations with coherent anomaly extrapolation [9, 10, 11, 12, 31] and could obtain estimates for the order parameter exponent with a few percent accuracy.

An interesting new class (**PCPD)**
appearing in annihilation fission systems, where there are two absorbing
states (one of them is fluctuating) without symmetry has just been investigated
numerically [29,
26,
32,
33,
36,
39,
41,43].
Field theoretical description and understanding of the reasons of this
class is still missing.

Recently reaction-diffusion models with production by
**triplets or quadruplets** and explicit
diffusion of particles have been found to exhibit novel types of
critical behavior in low dimensions. Mean-field calculations and
simulations in 1 and 2 dimansions were performed to determine the
phase transition behavior of such general types of systems as well as the
upper critical dimensions [38,
47].

The application of the basic absorbing state critical phenomena can be observed in SOC sandpile models or in certain surface growth models (see my page "Surface growth phenomena").

**First order transitions** have rarely
been seen in one dimension [53].
There is a hypothesis from Hinrichsen, that
they do not exists in one dimensional systems without extra symmetries,
conservation laws or long range interactions. First order transitions have
been found in the strong diffusion, mean-field like limit of the NEKIM model
[13]
and in case of the spin variables of the NEKIM model
[17,
18]
exhibiting Z_2 symmetry. ** In two dimensions ** hybrid
triplet and quadruplet reaction-diffusion (RD) models
[38]
or the Ising model coupled to a quadruplet [RD] field
[50] exhibit discontinuos transition for example.
A combination of first and second order transition is the so-called
**mixed-order or hybrid phase transition**.
This means that at the transition point the order parameter exhibits a jump,
in conjunction with critical phenomena related to it
[90]

By going beyond the site mean-field approximation it turned out that the
phase diagrams of reaction-diffusion models may contain other
transition points with non-trivial scaling behavior.
Cluster mean-field approximations and simulations in one and two dimensions
showed that the ** diffusion ** plays an
important role: it introduces a different critical point for strong
couplings [44] besides those obtained by site
mean-field solution at zero branching rate or by perturbative field theory.
The non-trivial critical point, appearing by low diffusion rates exhibits
the universal behavior of the transition of the 2A -> 3A, 2A -> 0 (PCPD)
model owing to the generation of the effective 2A -> 0 reaction via the
quick processes: 2A -> 3A -> 4A -> 0
[41,43].

Extensive simulations [46] suggest that PC class and more generally models with pair annihilation/coagulation reaction are a candidates for experimental observation. On the other hand spin-anisotropy, breaking the Z_2 symmetry can bring back DP type Griffiths phases as well as nonuniversal scaling [49]. Disorder can also be relevant in exclusion processes and lattice gases, causing different scaling behavior usually with slow dynamics [63].

It is known both for equilibrium and nonequilibrium systems
that the presence of **long-range interactions**
leads to different critical behavior compared to the universality classes
characteristic for systems with short-range interactions.
An alternative way of realizing long-range interactions is when the dynamical
process is defined on a network with long links, which connect distant sites.
These links can be fixed, i.e realize "quenched", long-range interactions.
Networks can also be composed on top of d-dimensional regular lattices by
additional long edges.

These arise e.g. in
sociophysics , or in the context of conductive properties
of linear polymers with crosslinks that connect remote monomers.
Putting the contact process (A -> 2A, A -> 0) onto such networks we have
provided numerical evidence that an absorbing phase transition
occurs at some finite value of the infection rate and
the corresponding dynamical critical exponents depend
on the underlying network.
Furthermore, the time dependent quantities exhibit log-periodic
oscillations in agreement with the discrete scale invariance
of the networks [54].
In case of **random connections **
rare regions, non-universal scaling (Griffiths phase) and generic
slow behavior may occur in these networks [57],
[59],[62],
[64],[67],
[68],[69,
[71],[72],
[73][74],
[77],[82],
[85].

Heterogeneities also play an important role in network **synchronization models**,
[87], which occur in the human brain
[88],
[r1], or in power-grids [83], [86].
Topological heterogeneity sometimes can be more relevant than interaction heterogeneity, altering the critical, universal
scaling behaior of the dynamical percolation of the Susceptible Infected Recovered epidemic model
[91].

For an overview of nonequilibrium universality classes see:
[42].

*June 9, 2021.*