ABSTRACT (original article): The antiferromagnetic q-state Potts model is perhaps the most canonical model for which the uniqueness threshold on the tree is not yet understood, largely because of the absence of monotonicities. Jonasson established the uniqueness threshold in the zero-temperature case, which corresponds to the q-colourings model. In the permissive case (where the temperature is positive), the Potts model has an extra parameter , which makes the task of analysing the uniqueness threshold even harder and much less is known. In this paper, we focus on the case q = 3 and give a detailed analysis of the Potts model on the tree by refining Jonasson’s approach. In particular, we establish the uniqueness threshold on the d-ary tree for all values of . When d ≥ 3, we show that the 3-state antiferromagnetic Potts model has uniqueness for all . The case is critical since it relates to the 3-colourings model on the binary tree , which has non-uniqueness. Nevertheless, we show that the Potts model has uniqueness for all on the binary tree. Both of these results are tight since it is known that uniqueness does not hold in the complementary regime. Our proof technique gives for general an analytical condition for proving uniqueness based on the two-step recursion on the tree, which we conjecture to be sufficient to establish the uniqueness threshold for all non-critical cases . CITATION (original article): Andreas Galanis, Leslie Ann Goldberg, Kuan Yang, “Uniqueness for the 3-state antiferromagnetic Potts model on the tree.” Electron. J. Probab. 23 1 - 43, 2018. https://doi.org/10.1214/18-EJP211
β∈(0,1)
d≥2
β≥1−3/(d+1)
d=2
(β=0)
β∈(0,1)
q>3
(q≠d+1)
We establish the uniqueness threshold for the 3-state antiferromagnetic Potts model on the infinite d-ary tree by tracking how boundary conditions influence the root via a carefully chosen ratio recursion, refining Jonasson’s approach from the zero-temperature colouring case to the positive-temperature setting with parameter .
We support the analytic argument with rigorous, machine-checked algebra: confirm symbolic derivatives used to prove monotonicity and contraction, and certify key inequality and non-existence claims by quantifier elimination (Mathematica Resolve) over the constrained parameter domains that arise in the proof.
β∈(0,1)
We support the analytic argument with rigorous, machine-checked algebra: confirm symbolic derivatives used to prove monotonicity and contraction, and certify key inequality and non-existence claims by quantifier elimination (Mathematica Resolve) over the constrained parameter domains that arise in the proof.
7.1 Lemma 2.11
7.1 Lemma 2.11
The following code checks that, for all d-tuples (,...,)∈ (α), it holds that ,,(,...,)< . The output is True, and the same is true when the first line changes to q = 4, d = 4.
(1)
p
(d)
p
Ex
q
h
c
1
c
2
β
*
(1)
p
(d)
p
1/d
α
In[]:=
q=3;d=3;b=1-q/(d+1);EX=Tuples[{1,alpha},q-1];G[vector_,colour_]:=1-(1-b)(vector[[colour]]-1)/(b+Sum[vector[[cc]],{cc,1,q-1}]);dTUPLES=Tuples[EX,d];L=Length[dTUPLES];UNIQ=True;For[l=1,l<=L&&UNIQ,l++,currenttuple=dTUPLES[[l]];For[k=1,k<=d,k++,vectorofchild[k]=currenttuple[[k]];];For[colour=1,colour<=q-1,colour++,g[colour]=Product[G[vectorofchild[k],colour],{k,1,d}];];h=1+(1-b)(1-g[1])/(b+Sum[g[c],{c,1,q-1}])/.{alpha->u^d};CHECK=Resolve[Exists[u,h>=u&&u>1]];If[CHECK==True,UNIQ=False];];Print[UNIQ]
True
7.2 Lemma 5.3
7.2 Lemma 5.3
Verify that the closed-form expressions for /∂β and /∂β derived in the paper are algebraically exact, so that all subsequent monotonicity-in-β arguments rest on certified identities rather than heuristic differentiation.
∂
f
u
∂
f
ℓ
Both of the queries in the following code give the output True. (Use on fresh kernel)
In[]:=
fu=(1-(1-b)y)^d/((1-(1-b)y)^d+2(1-(1-b)x)^(d/2)(1-(1-b)(1-x-y))^(d/2));bb=1-b;W=1-3y+bb((3y-1)+(1-x-2y)+x(2x+y-1)+y(x-y));lhs=D[fu,b];rhs=-fu^2(d(1-bbx)^(d/2)(1-bb(1-x-y))^(d/2)W/((1-bby)^(d+1)(1-bbx)(1-bb(1-x-y))));FullSimplify[lhs==rhs]fl=(1-(1-b)x)^d/((1-(1-b)x)^d+(1-(1-b)y)^d+(1-(1-b)(1-x-y))^d);lhs=D[fl,b];rhs=fl^2d((2x+y-1)(1-bb(1-x-y))^(d-1)+(x-y)(1-bby)^(d-1))/((1-bbx)^(d+1));FullSimplify[lhs==rhs]
Out[]=
True
Out[]=
True
7.3 Lemma 5.8
7.3 Lemma 5.8
Certify that the recursion fixed-point system on the binary tree () admits no solution outside the target marginal bounds, rigorously ruling out limiting configurations that would obstruct uniqueness.
d=2
Both of the queries in the following code give the output False.
In[]:=
fu=(1-(1-b)y)^d/((1-(1-b)y)^d+2(1-(1-b)x)^(d/2)(1-(1-b)(1-x-y))^(d/2));fl=(1-(1-b)x)^d/((1-(1-b)x)^d+(1-(1-b)y)^d+(1-(1-b)(1-x-y))^d);Resolve[Exists[{x,y,b},0<y<=1/3&&1106/2500<=x<1&&0<b<=1&&(fu/.{d->2})==x&&(fl/.{d->2})==y]]Resolve[Exists[{x,y,b},0<y<=460/2000&&1/3<=x<1&&0<b<=1&&(fu/.{d->2})==x&&(fl/.{d->2})==y]]
Out[]=
False
Out[]=
False
7.4 Lemma 5.15
7.4 Lemma 5.15
Verify the closed-form derivative of the upper-envelope drift with respect to y along the critical-parameter ray , confirming that is monotone decreasing in y wherever the contraction argument requires it.
g
u
x=my
g
u
The following code gives the output True.
In[]:=
fu=(1-(1-b)y)^d/((1-(1-b)y)^d+2(1-(1-b)x)^(d/2)(1-(1-b)(1-x-y))^(d/2));gu=fu-x/.{b->1-3/(d+1),x->my};A=(1-3y/(d+1))^d;B=(1-3(1-y(m+1))/(d+1))^(d/2)(1-3my/(d+1))^(d/2);W=AB/(A+2B)^2;lhs=D[gu,y];rhs=-m+(9dW/(3y(m+1)+d-2))(m(2my+y-1)/(1+d-3my)-(2my+y+d-1)/(1+d-3y));FullSimplify[lhs==rhs]
Out[]=
True
7.5 Lemma 5.16
7.5 Lemma 5.16
Verify the analogous derivative identity for the lower-envelope drift , confirming its monotone decrease in y and completing the pair of derivative certifications on which the envelope-contraction proof depends.
g
ℓ
The following code gives the output True.
In[]:=
fl=(1-(1-b)x)^d/((1-(1-b)x)^d+(1-(1-b)y)^d+(1-(1-b)(1-x-y))^d);gl=fl-y/.{b->1-3/(d+1),x->my};W=(m(2d-1)+d+1)((3y(m+1)+d-2)/(d+1))^d/((1+d-3my)(d-2+3y(1+m)))+(m-1)(d+1)(1-(3y)/(d+1))^d/((1+d-3y)(1+d-3my));lhs=D[gl,y];rhs=-3d(1-(3my)/(d+1))^dW/(((3y(m+1)+d-2)/(d+1))^d+(1-(3my)/(d+1))^d+(1-(3y)/(d+1))^d)^2-1;FullSimplify[lhs==rhs]
Out[]=
True
7.6 Lemma 5.18
7.6 Lemma 5.18
Confirm that the two explicit candidate points used in the case-split remain strictly inside the admissible simplex for all relevant values of the slope parameter
m, validating the geometric pre-condition for the derivative sign arguments of Lemmas 5.15 and 5.16.
(,)
x
i
y
i
m, validating the geometric pre-condition for the derivative sign arguments of Lemmas 5.15 and 5.16.
The following code outputs False and False.
In[]:=
y2=7/(10m+12);y1=y2+3/500;x2=my2;x1=my1;Resolve[Exists[m,157/80<=m<32&&(0>=y1||y1>=1-x1-y1||1-x1-y1>=1/3||1/3>=x1||x1>=1-y1)]]Resolve[Exists[m,32<=m&&(0>=y2||y2>=1-x2-y2||1-x2-y2>=1/3||1/3>=x2||x2>=1-y2)]]
Out[]=
False
Out[]=
False
7.7 Lemma 5.19
7.7 Lemma 5.19
Verify the closed-form expressions for /∂x and /∂y, confirming that the sign of each partial derivative is controlled by a simple polynomial factor — the foundation for proving directional monotonicity of in the drift bound.
∂
f
u
∂
f
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f
u
Both queries in the following code for the differentiation in (5.11) output True.
In[]:=
fu=(1-(1-b)y)^d/((1-(1-b)y)^d+2(1-(1-b)x)^(d/2)(1-(1-b)(1-x-y))^(d/2));bb=1-b;R=(1-bbx)^(d/2)(1-bb(1-x-y))^(d/2)/(1-bby)^d;xlhs=D[fu,x];xrhs=fu^2Rdbb^2(2x+y-1)/((1-bb(1-x-y))(1-bbx));ylhs=D[fu,y];yrhs=-fu^2Rdbb(3+bb(2x+y-2))/((1-bb(1-x-y))(1-bby));FullSimplify[xlhs==xrhs]FullSimplify[ylhs==yrhs]
Out[]=
True
Out[]=
True
Certify the uniform upper bound on the error term arising from the partial derivatives of , which is needed to reduce the key contraction inequality (5.15) to a form amenable to the final case analysis.
z≤1/24
f
u
The proof of (5.14). We consider two cases — when µ < 32 and when µ ≥ 32. The output is False in both cases.
In[]:=
lhs=(8-y)(2x+y-1)/((8-x)(2x+y+22));x1=7m/(10m+12)+3m/500;y1=7/(10m+12)+3/500;Resolve[Exists[m,1<m<32&&(lhs/.{x->x1,y->y1})>=1/24]]x2=7m/(10m+12);y2=7/(10m+12);Resolve[Exists[m,m>=32&&(lhs/.{x->x2,y->y2})>=1/24]]
Out[]=
False
Out[]=
False
Here is the code to show that Y is increasing in . The output is False.
β
*
In[]:=
Num=3+b(2x+y-2);Den=(1-b(1-x-y))(1-by);Der=D[Num/Den,b];Resolve[Exists[{d,x,y,b},d>=0&&0<b<=1/8&&0<y<1-x-y<1/3<x<1-y&&Der<0]]
Here is the code for Case 1. The output is False.
Here is the code for Case 2. The output is False.
7.8 Lemma 5.21
7.8 Lemma 5.21
The output of the following code, for the differentiation in (5.17), is True.
The following code establishes Facts 1, 2, and 3 for µ = 157/80. The output is False, 0, then True.
The following code establishes Facts 1, 2, and 3 for µ = 32. The output is False, 0, then True.
7.9 Lemma 5.23
7.9 Lemma 5.23
We first prove that h(23,µ) > 0 for µ ≥ 157/80. The output to both queries is False.
The output of the following code, for the differentiation in (5.19), is True.
7.10 Lemma 5.26
7.10 Lemma 5.26
The code checks that all of the desired inequalities are satisfied. The output is True.
7.11 Lemma 6.11
7.11 Lemma 6.11
7.12 Lemma 6.12
7.12 Lemma 6.12
The following code outputs True, therefore verifying the differentiation in (6.6).
The following code outputs True True, verifying the calculation for Step 3.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Uniqueness for the 3-state antiferromagnetic Potts model on the tree
by Andreas Galanis, Leslie Ann Goldberg, Kuan Yang
Wolfram Community, STAFF PICKS, March 6, 2026
https://community.wolfram.com/groups/-/m/t/3651601
by Andreas Galanis, Leslie Ann Goldberg, Kuan Yang
Wolfram Community, STAFF PICKS, March 6, 2026
https://community.wolfram.com/groups/-/m/t/3651601