Yinon Spinka
Office: MATX 1220
About me
I am currently a postdoc working in the field of probability under the supervision of
Omer Angel. I completed my PhD and master's at TelAviv University under the supervision of
Ron Peled.
Selected papers

Rigidity of proper colorings of $\mathbb{Z}^d$
Preprint
with Ron Peled
A proper $q$coloring of a domain in $\mathbb{Z}^d$ is a function assigning one of $q$ colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper $q$coloring uniformly at random, does the coloring typically exhibit longrange order? It has been known since the work of Dobrushin that no such ordering can arise when $q$ is large compared with $d$. We prove here that longrange order does arise for each $q$ when $d$ is sufficiently high, and further characterize all periodic maximalentropy Gibbs states for the model. Ordering is also shown to emerge in low dimensions if the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$, $d=d_1+d_2$ sufficiently high and $\mathbb{T}$ a cycle of even length. The results address questions going back to BerkerKadanoff (1980), Kotecký (1985) and SalasSokal (1997).

Longrange order in discrete spin systems
Preprint
with Ron Peled
We establish longrange order for discrete nearestneighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described threshold. The results characterize all periodic, maximalpressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T}$ is a cycle of even length. Applications to specific systems are discussed in detail and models for which new results are provided include the antiferromagnetic Potts model, Lipschitz height functions, and the hardcore, WidomRowlinson and beach models and their multitype extensions.

Macroscopic loops in the loop O(n) model at Nienhuis' critical point
JEMS
(Journal of the European Mathematical Society)
(to appear)
with Hugo DuminilCopin, Alexander Glazman, Ron Peled
The loop $O(n)$ model is a model for a random collection of nonintersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2n}}}$. For $0 \lt n\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x\lt x_c(n)$ from a regime with macroscopic loops when $x\ge x_c(n)$.
In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. This is the first instance in which a loop $O(n)$ model with $n\neq 1$ is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \ge 1$ and $0\lt x\le\frac{1}{\sqrt{n}}$. This property implies, using techniques recently developed for the randomcluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (boxcrossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$ and $n\in[1,2]$.

Finitary codings for spatial mixing Markov random fields
AOP
(Annals of Probability)
It has been shown by van den Berg and Steif that the subcritical and critical Ising model on $\mathbb{Z}^d$ is a finitary factor of an i.i.d. process (ffiid), whereas the supercritical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with powerlaw tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical twodimensional models. Using a result of the author, we deduce that strong spatial mixing also implies ffiid with stretchedexponential tails from a finitevalued i.i.d. process.
We give several applications to models such as the Potts model, proper colorings, the hardcore model, the WidomRowlinson model and the beach model. For instance, for the ferromagnetic $q$state Potts model on $\mathbb{Z}^d$ at inverse temperature $\beta$, we show that it is ffiid with exponential tails if $\beta$ is sufficiently small, it is ffiid if $\beta < \beta_c(q,d)$, it is not ffiid if $\beta > \beta_c(q,d)$ and, when $d=2$ and $\beta=\beta_c(q,d)$, it is ffiid if and only if $q \le 4$.

A short proof of the discontinuity of phase transition in the planar randomcluster model with $q>4$
CMP
(Communications in Mathematical Physics)
with Gourab Ray
The goal of this paper is to provide a short proof of the discontinuity of phase transition for the randomcluster model on the square lattice with parameter $q>4$. This result was recently shown via the socalled Bethe ansatz for the sixvertex model. Our proof also exploits the connection to the sixvertex model, but does not rely on the Bethe ansatz. Our argument is soft and only uses very basic properties of the randomcluster model (for example, we do not need the RussoSeymourWelsh theory).
Full list of papers

Longrange order in discrete spin systems
Preprint
with Ron Peled
We establish longrange order for discrete nearestneighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described threshold. The results characterize all periodic, maximalpressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T}$ is a cycle of even length. Applications to specific systems are discussed in detail and models for which new results are provided include the antiferromagnetic Potts model, Lipschitz height functions, and the hardcore, WidomRowlinson and beach models and their multitype extensions.

Proper 3colorings of $\mathbb{Z}^2$ are Bernoulli
Preprint
with Gourab Ray
We consider the unique measure of maximal entropy for proper 3colorings of $\mathbb{Z}^2$, or equivalently, the socalled zeroslope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translationequivariant function of independent and identically distributed random variables placed on $\mathbb{Z}^2$. Along the way, we obtain various estimates on the mixing properties of this measure.

Three lectures on random proper colorings of $\mathbb{Z}^d$
Preprint
with Ron Peled
A proper $q$coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$colorings of a large discrete cube in the integer lattice $\mathbb{Z}^d$. Does the random coloring obtained exhibit any largescale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension $d$ and the number of colors $q$. The questions are motivated by statistical physics (antiferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.

Geometric random graphs on circles
Preprint
with Omer Angel
Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference $L$, and find a surprising dependency of behavior on the rationality of $L$.

Finitary isomorphisms of renewal point processes and continuoustime regenerative processes
Preprint
We show that a large class of stationary continuoustime regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo and of Kosloff and Soo.

Finitary codings for gradient models and a new graphical representation for the sixvertex model
Preprint
with Gourab Ray
It is known that the Ising model on $\mathbb {Z}^d$ at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction for being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volumeorder large deviation estimate for the energy. A similar result is shown for the Potts model.
A result in the same spirit is also shown for the sixvertex model, which is itself the gradient of a height function, with parameter $c \gtrapprox 6.4$. We show that the gradient of the height function is not a finitary factor of an i.i.d. process, but that its "Laplacian" is. For this, we introduce a coupling between the sixvertex model with $c\ge 2$ and a new graphical representation of it, reminiscent of the EdwardsSokal coupling between the Potts and randomcluster models. We believe that this graphical representation may be of independent interest and could serve as a tool in further understanding of the sixvertex model.
To provide further support for the ubiquity of this type of phenomenon, we also prove an analogous result for the socalled beach model.
The tools and techniques used in this paper are probabilistic in nature. The heart of the argument is to devise a suitable tree structure on the clusters of the underlying percolation process (associated to the graphical representation of the given model), which can be revealed piecebypiece via exploration.

Markov chains with exponential return times are finitary
ETDS
(Ergodic Theory and Dynamical Systems)
with Omer Angel
Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an i.i.d. process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of $\mathbb{Z}$ is a finitary factor of an i.i.d. process.

A short proof of the discontinuity of phase transition in the planar randomcluster model with $q>4$
CMP
(Communications in Mathematical Physics)
with Gourab Ray
The goal of this paper is to provide a short proof of the discontinuity of phase transition for the randomcluster model on the square lattice with parameter $q>4$. This result was recently shown via the socalled Bethe ansatz for the sixvertex model. Our proof also exploits the connection to the sixvertex model, but does not rely on the Bethe ansatz. Our argument is soft and only uses very basic properties of the randomcluster model (for example, we do not need the RussoSeymourWelsh theory).

Mixing properties of colorings of the $\mathbb{Z}^d$ lattice
CPC
(Combinatorics, Probability and Computing)
with Noga Alon, Raimundo Briceño, Nishant Chandgotia, Alexander Magazinov
We study and classify proper $q$colorings of the $\mathbb Z^d$ lattice, identifying three regimes where different combinatorial behavior holds: (1) When $q\le d+1$, there exist frozen colorings, that is, proper $q$colorings of $\mathbb Z^d$ which cannot be modified on any finite subset. (2) We prove a strong listcoloring property which implies that, when $q\ge d+2$, any proper $q$coloring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper $q$coloring of the entire box. (3) When $q\geq 2d+1$, the latter holds for any $n \ge 1$. Consequently, we classify the space of proper $q$colorings of the $\mathbb Z^d$ lattice by their mixing properties.

Pairwise optimal coupling of multiple random variables
Preprint
with Omer Angel
We generalize the optimal coupling theorem to multiple random variables: Given a collection of random variables, it is possible to couple all of them so that any two differ with probability comparable to the totalvariation distance between them. In a number of cases we show that the disagreement probability we achieve is the best possible. The proofs of sharpness rely on new results in extremal combinatorics, which may be of independent interest.

Finitely dependent processes are finitary
AOP
(Annals of Probability)
We show that any finitely dependent invariant process on a transitive amenable graph is a finitary factor of an i.i.d. process. With an additional assumption on the geometry of the graph, namely that no two balls with different centers are identical, we further show that the i.i.d. process may be taken to have entropy arbitrarily close to that of the finitely dependent process. As an application, we give an affirmative answer to a question of Holroyd.

Rigidity of proper colorings of $\mathbb{Z}^d$
Preprint
with Ron Peled
A proper $q$coloring of a domain in $\mathbb{Z}^d$ is a function assigning one of $q$ colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper $q$coloring uniformly at random, does the coloring typically exhibit longrange order? It has been known since the work of Dobrushin that no such ordering can arise when $q$ is large compared with $d$. We prove here that longrange order does arise for each $q$ when $d$ is sufficiently high, and further characterize all periodic maximalentropy Gibbs states for the model. Ordering is also shown to emerge in low dimensions if the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$, $d=d_1+d_2$ sufficiently high and $\mathbb{T}$ a cycle of even length. The results address questions going back to BerkerKadanoff (1980), Kotecký (1985) and SalasSokal (1997).

Finitary codings for the randomcluster model and other infiniterange monotone models
Preprint
with Matan Harel
A random field $X = (X_v)_{v \in G}$ on a quasitransitive graph $G$ is a factor of i.i.d. if it can be written as $X=\varphi(Y)$ for some i.i.d. process $Y= (Y_v)_{v \in G}$ and equivariant map $\varphi$. Such a map, also called a coding, is finitary if, for every vertex $v \in G$, there exists a finite (but random) set $U \subset G$ such that $X_v$ is determined by $\{Y_u\}_{u \in U}$. We construct a coding for the randomcluster model on $G$, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of HäggströmJonassonLyons. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model.
Our methods are probabilistic in nature, and at their heart lies the use of couplingfromthepast for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the randomcluster and Potts models, we describe two further applications  the loop $O(n)$ model and longrange Ising models. In the case of $G = \mathbb{Z}^d$, we also construct finitary, translationequivariant codings using a finitevalued i.i.d. process $Y$. To do this, we extend a mixingtime result of MartinelliOlivieri to infiniterange monotone models on quasitransitive graphs of subexponential growth.

Finitary codings for spatial mixing Markov random fields
AOP
(Annals of Probability)
It has been shown by van den Berg and Steif that the subcritical and critical Ising model on $\mathbb{Z}^d$ is a finitary factor of an i.i.d. process (ffiid), whereas the supercritical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with powerlaw tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical twodimensional models. Using a result of the author, we deduce that strong spatial mixing also implies ffiid with stretchedexponential tails from a finitevalued i.i.d. process.
We give several applications to models such as the Potts model, proper colorings, the hardcore model, the WidomRowlinson model and the beach model. For instance, for the ferromagnetic $q$state Potts model on $\mathbb{Z}^d$ at inverse temperature $\beta$, we show that it is ffiid with exponential tails if $\beta$ is sufficiently small, it is ffiid if $\beta < \beta_c(q,d)$, it is not ffiid if $\beta > \beta_c(q,d)$ and, when $d=2$ and $\beta=\beta_c(q,d)$, it is ffiid if and only if $q \le 4$.

Finitary coding for the subcritical Ising model with finite expected coding volume
EJP
(Electronic Journal of Probability)
It has been shown by van den Berg and Steif that the subcritical Ising model on $\mathbb{Z}^d$ is a finitary factor of a finitevalued i.i.d. process. We strengthen this by showing that the factor map can be made to have finite expected coding volume (in fact, stretchedexponential tails), answering a question of van den Berg and Steif. The result holds at any temperature above the critical temperature. An analogous result holds for Markov random fields satisfying a highnoise assumption and for proper colorings with a large number of colors.

A condition for longrange order in discrete spin systems with application to the antiferromagnetic Potts model
Preprint
with Ron Peled
We give a general condition for a discrete spin system with nearestneighbor interactions on the $\mathbb{Z}^d$ lattice to exhibit longrange order. The condition is applicable to systems with residual entropy in which the longrange order is entropically driven. As a main example we consider the antiferromagnetic $q$state Potts model and rigorously prove the existence of a broken sublattice symmetry phase at low temperature and high dimension  a new result for $q\ge 4$. As further examples, we prove the existence of an ordered phase in a clock model with hard constraints and extend the known regime of the demixed phase in the lattice WidomRowlinson model.

Lectures on the spin and loop O(n) models
Sojourns in Probability Theory and Statistical Physics
(celebrating Charles Newman's 70th birthday)
with Ron Peled
The classical spin $O(n)$ model is a model on a $d$dimensional lattice in which a vector on the $(n1)$dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model ($n=1$), the XY model ($n=2$) and the Heisenberg model ($n=3$). We discuss questions of longrange order and decay of correlations in the spin $O(n)$ model for different combinations of the lattice dimension $d$ and the number of spin components $n$.
The loop $O(n)$ model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight $n\ge0$ and an edge weight $x\ge 0$. Special cases include selfavoiding walk ($n=0$), the Ising model ($n=1$), critical percolation ($n=x=1$), dimer model ($n=1,x=\infty$), proper $4$coloring ($n=2, x=\infty)$, integervalued ($n=2$) and treevalued (integer $n>=3$) Lipschitz functions and the hard hexagon model ($n=\infty$). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin $O(n)$ model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.

Macroscopic loops in the loop O(n) model at Nienhuis' critical point
JEMS
(Journal of the European Mathematical Society)
(to appear)
with Hugo DuminilCopin, Alexander Glazman, Ron Peled
The loop $O(n)$ model is a model for a random collection of nonintersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2n}}}$. For $0 \lt n\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x\lt x_c(n)$ from a regime with macroscopic loops when $x\ge x_c(n)$.
In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. This is the first instance in which a loop $O(n)$ model with $n\neq 1$ is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \ge 1$ and $0\lt x\le\frac{1}{\sqrt{n}}$. This property implies, using techniques recently developed for the randomcluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (boxcrossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$ and $n\in[1,2]$.

The growth constant of odd cutsets in high dimensions
CPC
(Combinatorics, Probability and Computing)
with Ohad Feldheim
A cutset is a nonempty finite subset of $\mathbb{Z}^d$ which is both connected and coconnected. A cutset is odd if its vertex boundary lies in the odd bipartition class of $\mathbb{Z}^d$. Peled suggested that the number of odd cutsets which contain the origin and have $n$ boundary edges may be of order $e^{\Theta(n/d)}$ as $d \to \infty$, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel to be of order $d^{\Theta(n/d)}$. In this paper, we verify this by showing that the number of such odd cutsets is $(2+o(1))^{n/2d}$.

Longrange order in the 3state antiferromagnetic Potts model in high dimensions
JEMS
(Journal of the European Mathematical Society)
with Ohad Feldheim
We prove the existence of longrange order for the 3state Potts antiferromagnet at low temperature on $\mathbb{Z}^d$ for sufficiently large $d$. In particular, we show the existence of six extremal and ergodic infinitevolume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the highdimensional case of the Kotecký conjecture.

On the converse of Talagrand's influence inequality
Preprint
with Saleet Klein, Amit Levi, Muli Safra, Clara Shikhelman
In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $0\lt a_1,a_2,\ldots,a_n\le 1$ such that $\sum_{j=1}^n a_j/(1\log a_j)\ge C$ for some constant $C>0$, it is possible to find a roughly balanced Boolean function $f$ such that $\textrm{Inf}_j[f] \lt a_j$ for every $1 \le j \le n$.

Exponential decay of loop lengths in the loop O(n) model with large n
CMP
(Communications in Mathematical Physics)
with Hugo DuminilCopin, Ron Peled, Wojciech Samotij
The loop $O(n)$ model is a model for a random collection of nonintersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ models exhibit exponential decay of correlations when $n>2$. We verify this for the loop $O(n)$ model with large parameter $n$, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight $x$. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when $nx^6$ is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when $nx^6$ is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

Random walk with longrange constraints
EJP
(Electronic Journal of Probability)
with Ron Peled
We consider a model of a random height function with longrange constraints on a discrete segment. This model was suggested by Benjamini, Yadin and Yehudayoff and is a generalization of simple random walk. The random function is uniformly sampled from all graph homomorphisms from the graph $P_{n,d}$ to the integers $\mathbb{Z}$, where the graph $P_{n,d}$ is the discrete segment $\{0,1,..., n\}$ with edges between vertices of different parity whose distance is at most $2d+1$. Such a graph homomorphism can be viewed as a height function whose values change by exactly one along edges of the graph $P_{n,d}$. We also consider a similarly defined model on the discrete torus.
Benjamini, Yadin and Yehudayoff conjectured that this model undergoes a phase transition from a delocalized to a localized phase when $d$ grows beyond a threshold $c*\log(n)$. We establish this conjecture with the precise threshold $\log_2(n)$. Our results provide information on the typical range and variance of the height function for every given pair of $n$ and $d$, including the critical case when $d\log_2(n)$ tends to a constant.
In addition, we identify the local limit of the model, when $d$ is constant and $n$ tends to infinity, as an explicitly defined Markov chain.
Coauthors
Some Pictures
The loop O(n) model
n=0.5, x=0.6
n=2.4, x=1
The spin O(n) model
The Ising model (n=1)
beta=0.4
beta=0.4407 (critical)
beta=0.5
beta=0.5 with Dobrushin b.c.
The XY model (n=2)
beta=1
beta=1.12 (critical)
beta=1.5
beta=3
The Heisenberg model (n=3)
beta=2
beta=10
3colorings (the antiferromagnetic Potts model)
2D, beta=4
slab in 3D, beta=4
Random walk with longrange constraints
discrete (integer valued)
continuous (real valued)