arXiv subjects

Paulina Smolarova

Publications and source records attributed to Paulina Smolarova.

4 recordsLinked to original sources

arXiv2026

Fast Mixing for Low-Temperature Potts Models via Poisson Trees

The $q$-state ferromagnetic Potts model on a graph $G$ is a probability distribution on all $q$-colourings of $G$ that favours many monochromatic edges. Approximate sampling from the Potts model is a central problem in the study of spin systems on sparse graphs, especially in the low-temperature regime, where the model strongly favours ordered configurations, often creating bottlenecks that make Markov-chain sampling inefficient or difficult to analyse. We focus on the sparse random graph $G(n,d/n)$. The local neighbourhoods of $G(n,d/n)$ are tree-like, but the relevant underlying graph is a Poisson Galton-Watson tree. This motivates the study of Glauber dynamics for the low-temperature Potts model on such trees with monochromatic boundary conditions. The Poisson setting introduces difficulties absent from the regular case: degrees fluctuate, long induced paths may appear, and branches can terminate before reaching the boundary. As a result, the effect of the monochromatic boundary at the leaves is much less uniform. Our main result shows near-linear mixing for the Glauber dynamics on Poisson trees with monochromatic boundary conditions. This extends the corresponding regular-tree results of Martinelli, Sinclair, and Weitz (SODA 2004) and of Blanca, Chen, Stefankovič, and Vigoda (RANDOM 2021) to the irregular trees arising from sparse random graphs. Our proof introduces an adaptive block decomposition of the tree, built around regions containing large regular subtrees, and combines it with correlation-decay estimates and functional-inequality arguments. We also obtain a near-linear-time approximate sampling algorithm for the Potts model on $G(n,d/n)$ at all temperatures, speeding up the best previous algorithm of Galanis, Goldberg, and Smolarova (ICALP 2025). The main new ingredient is a refined analysis of the low-temperature regime, building on the Poisson tree result.

math.PR↗

arXiv2026

Low-temperature Sampling on Sparse Random Graphs

We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from the ferromagnetic $q$-state Potts model. Low-temperature sampling is computationally intractable for general graphs, but recent advances based on the polymer method have made significant progress for graph families that exhibit certain expansion properties that reinforce the correlations, including for example expanders, lattices and dense graphs. One of the most natural graph classes that has so far escaped this algorithmic framework is the class of sparse Erdős-Rényi random graphs whose expansion only manifests for sufficiently large subsets of vertices; small sets of vertices on the other hand have vanishing expansion which makes them behave independently from the bulk of the graph and therefore weakens the correlations. At a more technical level, the expansion of small sets is crucial for establishing the Kotecky-Priess condition which underpins the applicability of the framework. Our main contribution is to develop the polymer method in the low-temperature regime for sparse random graphs. As our running example, we use the Potts and random-cluster models on $G(n,d/n)$ for $d=Θ(1)$, where we show a polynomial-time sampling algorithm for all sufficiently large $q$ and $d$, at all temperatures. Our approach applies more generally for models that are monotone. Key to our result is a simple polymer definition that blends easily with the connectivity properties of the graph and allows us to show that polymers have size at most $O(\log n)$.

cs.DM↗

arXiv2026

Planting and MCMC Sampling from the Potts model

We consider the problem of sampling from the ferromagnetic $q$-state Potts model on the random $d$-regular graph with parameter $β>0$. A key difficulty that arises in sampling from the model is the existence of a metastability window $(β_u,β_u')$ where the distribution has two competing modes, the so-called disordered and ordered phases, causing MCMC-based algorithms to be slow mixing from worst-case initialisations. To this end, Helmuth, Jenssen and Perkins designed a sampling algorithm that works for all $β$ when $q$ is large, using cluster expansion methods; more recently, their analysis technique has been adapted to show that random-cluster dynamics mixes fast when initialised more judiciously. However, a bottleneck behind cluster-expansion arguments is that they inherently only work for large $q$, whereas it is widely conjectured that sampling is possible for all $q,d\geq 3$. The only result so far that applies to general $q,d\geq 3$ is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast for $β<β_u$. For $β>β_u$, certain correlation phenomena emerge because of the metastability which have been hard to handle, especially for small $q$ and $d$. Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold $β_u$. We use planting as the main tool in our proofs, and combine it with the analysis of random-cluster dynamics. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers $q,d\geq 3$ beyond the uniqueness threshold $β_u$; our analysis works all the way up to the threshold $β_c\in (β_u,β_u')$ where the dominant mode switches from disordered to ordered. We also obtain an algorithm in the ordered regime $β>β_c$ that refines significantly the range of $q,d$.

math.PR↗

arXiv2023

Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics

We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $β>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $Δ$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $β$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $Δ$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $β$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.

math.PR↗