Siamak Taati — Page 2

arXiv2011

First-order transition in Potts models with "invisible' states: Rigorous proofs

In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475, arXiv:1012.4254], a class of Potts models with "invisible" states was introduced, for which the authors argued by numerical arguments and by a mean-field analysis that a first-order transition occurs. Here we show that the existence of this first-order transition can be proven rigorously, by relatively minor adaptations of existing proofs for ordinary Potts models. In our argument we present a random-cluster representation for the model, which might be of independent interest.

cond-mat.stat-mech↗

arXiv2010

Graph Coloring and Function Simulation

We prove that every partial function with finite domain and range can be effectively simulated through sequential colorings of graphs. Namely, we show that given a finite set $S=\{0,1,\ldots,m-1\}$ and a number $n \geq \max\{m,3\}$, any partial function $φ:S^{^p} \to S^{^q}$ (i.e. it may not be defined on some elements of its domain $S^{^p}$) can be effectively (i.e. in polynomial time) transformed to a simple graph $\matr{G}_{_{φ,n}}$ along with three sets of specified vertices $$X = \{x_{_{0}},x_{_{1}},\ldots,x_{_{p-1}}\}, \ \ Y = \{y_{_{0}},y_{_{1}},\ldots,y_{_{q-1}}\}, \ \ R = \{\Kv{0},\Kv{1},\ldots,\Kv{n-1}\},$$ such that any assignment $σ_{_{0}}: X \cup R \to \{0,1,\ldots,n-1\} $ with $σ_{_{0}}(\Kv{i})=i$ for all $0 \leq i < n$, is {\it uniquely} and {\it effectively} extendable to a proper $n$-coloring $σ$ of $\matr{G}_{_{φ,n}}$ for which we have $$φ(σ(x_{_{0}}),σ(x_{_{1}}),\ldots,σ(x_{_{p-1}}))=(σ(y_{_{0}}),σ(y_{_{1}}),\ldots,σ(y_{_{q-1}})),$$ unless $(σ(x_{_{0}}),σ(x_{_{1}}),\ldots,σ(x_{_{p-1}}))$ is not in the domain of $φ$ (in which case $σ_{_{0}}$ has no extension to a proper $n$-coloring of $\matr{G}_{_{φ,n}}$).

math.CO↗