  The Hubbard model is a longstanding problem in the theory of strongly
correlated electrons and a very active one in the experiments with ultracold
fermionic atoms. Motivated by current and prospective quantum simulations, we
apply a two-dimensional tensor network, an infinite projected entangled pair
state, evolved in imaginary time by the neighborhood tensor update algorithm
working directly in the thermodynamic limit. With U(1)xU(1) symmetry and the
bond dimensions up to 29, we generate thermal states down to the temperature of
0.17 times the hopping rate. We obtain results for spin and charge correlators,
unaffected by boundary effects. The spin correlators, measurable in prospective
ultracold atoms experiments attempting to approach the thermodynamic limit,
provide evidence of disruption of the antiferromagnetic background with mobile
holes in a slightly doped Hubbard model. The charge correlators reveal the
presence of hole-doublon pairs near half filling and signatures of hole-hole
repulsion on doping. We also obtain specific heat in the slightly doped regime.
