I don’t know about walkable but poor neighbourhoods full of recent immigrants have cheap supermarkets with fresh produce in the US. The poor neighbourhoods with no fresh produce don’t have it because the people living there don’t want it.
https://web.stanford.edu/~diamondr/AllcottDiamondDube_FoodDe...
> THE GEOGRAPHY OF POVERTY AND NUTRITION: FOOD DESERTS AND FOOD CHOICES ACROSS THE UNITED STATES
> We study the causes of “nutritional inequality”: why the wealthy tend to eat more healthfully than the poor in the U.S. Using event study designs exploiting supermarket entry and households' moves to healthier neighborhoods, we reject that neighborhood environments have meaningful effects on healthy eating. Using a structural demand model, we find that exposing low-income households to the same availability and prices experienced by high-income households reduces nutritional inequality by only 9%, while the remaining 91% is driven by differences in demand. These findings contrast with discussions of nutritional inequality that emphasize supply-side factors such as food deserts.